List

USAPhO Problems and Solutions

The United States selection, run by AAPT since 1986. Thousands sit the F=ma exam; around 400 go on to the USAPhO, six problems in three hours, and twenty are picked for the team. Every sitting AAPT publishes is here, from F=ma to the training camp.

2026

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Apparent transverse speed from a moving light source*not published
  • 2Scaling of a domino wave's speed with size*not published
  • 3Velocity components from a curved trajectory*not published
  • 4Distance between two balls when one lands*not published
  • 5Radius of the curve traced by dropped sand*not published
  • 6Scaling of impact force with projectile speed*not published
  • 7Stability of the Lagrange L2 point*not published
  • 8Maximum height of a puck riding up a moving bump*not published
  • 9Collisions needed for a bounced ball to escape Earth*not published
  • 10Comparing surface gravity of a half-ball and cylinders*not published
  • 11Track speed when a bead reaches a given tick mark*not published
  • 12Tow rope tension for a wake surfer at constant speed*not published
  • 13Total stopping time for a block bouncing between springs*not published
  • 14Period of a small oscillation in a rigid-rod swing*not published
  • 15Dependence of a rolling ball's final speed on friction*not published
  • 16Locating a puck's second bounce inside a triangular rack*not published
  • 17Speed of a sliding ladder's foot at a given position*not published
  • 18Period ratio for a three-spring oscillator*not published
  • 19Oscillation frequency of a pendulum on a sliding box*not published
  • 20Motor motion that keeps a pivoted rod's angle fixed*not published
  • 21Minimum jumps to escape a rotating merry-go-round*not published
  • 22Angular acceleration of a braking tricycle on ice*not published
  • 23Moment of inertia of a fractal hexagon pattern*not published
  • 24Maximum speed for air to stay attached to a cylinder*not published
  • 25Fastest-draining vase shape*not published
USAPhO
Problems6 problemsProblemsSolutions
  • 1A1. 404 Vision Not Foundnot published
  • 2A2. Straight on a Conenot published
  • 3A3. Storm Chasernot published
  • 4B1. Radiant Bubblesnot published
  • 5B2. Say Cheese... It's a Conicnot published
  • 6B3. The Trojan Cylindernot published

2025

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Position plots consistent with constant speed*not published
  • 2Final velocity after three disks collide simultaneously*not published
  • 3Final speed after three disks collide in sequence*not published
  • 4Acceleration of a cat pursuing a mouse*not published
  • 5Friction pairs that keep stacked cylinders stable*not published
  • 6Landing distance of a ball launched from different heights*not published
  • 7Total energy equation for a walking linkage*not published
  • 8Apparent spin of a top on a counter-rotating plate*not published
  • 9Acceleration extrema of a mass on nested rotating circles*not published
  • 10Maximum energy transfer from a heavy to a light object*not published
  • 11Maximum energy transfer from a light to a heavy object*not published
  • 12Maximum swing angle after a clay ball hits a pendulum*not published
  • 13Speed and spin graphs of a slipping, then rolling ball*not published
  • 14Comparing final speeds for balls that stop slipping*not published
  • 15Relating angular velocities after an impulse on a double rod*not published
  • 16Evolution of two connected soap bubbles*not published
  • 17Long-term path in an inverse-square-like potential*not published
  • 18Period of oscillation in a saddle-shaped potential*not published
  • 19Wind turbine power scaling with height*not published
  • 20Timing of a tossed spanner's closest return to an astronaut*not published
  • 21Speed difference between two pipe sections*not published
  • 22Height difference between two manometer tubes*not published
  • 23Percent error in a measured spring constant*not published
  • 24Return time for a bead impulsed on a tilted rod*not published
  • 25Final speed of a puck sliding off a moving prism*not published
USAPhO
Problems6 problemsProblemsSolutions
  • 1A1. Shake Itnot published
  • 2A2. Black Tidesnot published
  • 3A3. Bitter and Magneticnot published
  • 4B1. Scroll 'n' Rollnot published
  • 5B2. Where's the Kaboom?not published
  • 6B3. Locked and Modednot published

2024

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Height of hoops an arrow passes through*not published
  • 2Angular velocity for a rider's feet to lift off*not published
  • 3Tension and compression in a truss bridge*not published
  • 4Kinetic energy graph of a bouncing elastic ball*not published
  • 5Scale reading as a block slides down a ramp on it*not published
  • 6Water dripped from a swinging leaky bucket*not published
  • 7Velocity versus position from a velocity versus time graph*not published
  • 8Speed of a sliding rod's midpoint*not published
  • 9Stopping distance on a road that turns to ice*not published
  • 10Equilibrium radius of a block spun between two springs*not published
  • 11Force to pull apart a pressure-sealed sphere*not published
  • 12Torque on a probe from three asteroids*not published
  • 13Tension in a string as blocks are released, versus separation*not published
  • 14Angle traveled by a bead before matching its hoop's speed*not published
  • 15Constructing a dimensionless number for viscous drops*not published
  • 16Maximum landing distance released from a swinging rope*not published
  • 17Rope displacement between two equilibria with springs*not published
  • 18Delta-v to reach a larger elliptical orbit*not published
  • 19Oscillation frequency of an unbalanced spoked wheel*not published
  • 20Oscillation period of a spring-braced hinged square*not published
  • 21Force to push water out of a tapering syringe*not published
  • 22Gravity jump across a thin spherical shell*not published
  • 23Steady bounce height off a rising paddle*not published
  • 24Scaling of displacement at half terminal velocity in fluid*not published
  • 25Acceleration of a yo-yo while unwinding*not published
USAPhO
Problems6 problemsProblemsSolutions
  • 1A1. Ping Pongnot published
  • 2A2. Stellar Stabilitynot published
  • 3A3. Tilt Shiftnot published
  • 4B1. The Muon Shotnot published
  • 5B2. Solid Heatnot published
  • 6B3. Quality Questnot published

2023

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Average velocity of a bead going around a hoop*not published
  • 2Direction of acceleration at the bottom of a pendulum swing*not published
  • 3Vertical velocity graph of a bouncing soccer ball*not published
  • 4Work to drag a box slowly up an incline*not published
  • 5Acceleration of masses in a pulley system*not published
  • 6Height difference in a swinging conical pendulum*not published
  • 7Minimum friction to keep stacked boxes from sliding*not published
  • 8Speed ratio of two orbiting stars*not published
  • 9What loses energy as a balloon rises to the ceiling*not published
  • 10Aiming angle for an arrow to hit a target*not published
  • 11Maximum height for a straight-up throw versus a sideways one*not published
  • 12Tension change in a string after cutting another*not published
  • 13Sliding versus lifting off for a box pushed at an angle*not published
  • 14Weight fraction on one leg of a three-rod stand*not published
  • 15Enclosed mass from a galaxy's rotation curve*not published
  • 16Timing of a pendulum bouncing off a wall*not published
  • 17Ranking moments of inertia of three physical pendulums*not published
  • 18Ranking measured weights of bagged rocks in water*not published
  • 19Uncertainty in g from a pendulum experiment*not published
  • 20Velocity components curve of a swinging pendulum*not published
  • 21Return time of a mass sliding inside a ring*not published
  • 22Ranking effective spring constants from three spring combinations*not published
  • 23Distance ratio from a parallax angle shift*not published
  • 24Speed change bouncing off an oscillating plate*not published
  • 25Horizontal distance after a ski jump ramp*not published
USAPhO
Problems6 problemsProblemsSolutions
  • 1A1. Circus Actnot published
  • 2A2. Time is a Flat Circlenot published
  • 3A3. The Motive Power of Icenot published
  • 4B1. Electric Roulettenot published
  • 5B2. Fast and Furiousnot published
  • 6B3. Starry Messengersnot published

2022

USA

F=ma
Exam A25 problemsProblemsSolutions
  • 1Maximum height from height at half speed*not published
  • 2Speed hitting an obstacle at a higher initial speed*not published
  • 3Percent kinetic energy lost in an inelastic collision*not published
  • 4Trajectory after a pendulum string breaks*not published
  • 5Total kinetic energy of a rolling ball*not published
  • 6Maximum tension in a pendulum released horizontally*not published
  • 7Functional form from a measurement graph*not published
  • 8Friction force from the ground on a wedge*not published
  • 9Comparing forces to raise a mass at velocity and acceleration*not published
  • 10Oscillation period of a rod hung by two strings*not published
  • 11Minimum launch speed to reach a neighboring planet*not published
  • 12Force ratio on masses from a two-radius pulley*not published
  • 13Minimum force to tip a locked laptop*not published
  • 14True statements about a block sliding in a bowl*not published
  • 15Optimal angle for a projectile to hit a wall highest*not published
  • 16Minimum friction for a pushed pencil to roll*not published
  • 17What must be equal between two halves of a spinning rod*not published
  • 18Water height when a cork begins to float*not published
  • 19Hydrostatic force on a submerged toy with sealed legs*not published
  • 20Wire shape with zero normal force on a bead*not published
  • 21Cork height in a damped fluid during elevator motion*not published
  • 22Downward acceleration of a block on two wedges*not published
  • 23Comparing fall times with linear and quadratic drag*not published
  • 24Orbit area from a satellite's speed at apoapsis*not published
  • 25Minimum friction for a rubber band on a cylinder*not published
Exam B25 problemsProblemsSolutions
  • 26Distance between bounce points on a 45-degree slope*not published
  • 27Moments of inertia of a four-mass wheel about different axes*not published
  • 28Balancing mass after moving a seesaw's fulcrum*not published
  • 29Shape formed by firework shells over time*not published
  • 30Average speed of a round trip with a river current*not published
  • 31Momentum of a heavier block pushed the same way*not published
  • 32Scale reading while pulley masses are falling*not published
  • 33Launch angle for a ball to bounce twice and return*not published
  • 34Minimum force to slide a box pushed at an angle*not published
  • 35Force to stretch a combination of three springs*not published
  • 36Pressure force on a cup with a tied floating block*not published
  • 37Minimum friction to hold two roped bricks on a slope*not published
  • 38Maximum height of a ballistic pendulum with rotation*not published
  • 39Surface area and volume after a cylinder splits into droplets*not published
  • 40Which thrust direction keeps the same faster orbit*not published
  • 41Effective point of a normal force on an inclined block*not published
  • 42Timing between a puck's wall collisions in a sliding box*not published
  • 43Oscillation frequency of a rod on two springs*not published
  • 44Rotational oscillation frequency of a rod on two springs*not published
  • 45Acceleration of a hollow sphere on an accelerating plane*not published
  • 46Comparing uncertainty in a rectangle's area and perimeter*not published
  • 47Launch angles for a projectile to never hit its planet*not published
  • 48Velocity-acceleration curve of simple harmonic motion*not published
  • 49Kinetic energy gained sliding down a moving ramp*not published
  • 50Collision time scaling for heavier attracting masses*not published
USAPhO
Problems6 problemsProblemsSolutions
  • 1A1. Moment of Claritynot published
  • 2A2. Death Metalnot published
  • 3A3. Rainbow Roadnot published
  • 4B1. Virial Realitynot published
  • 5B2. Broken Vasenot published
  • 6B3. Time Crystalnot published
Training Camp
Theoretical3 problemsProblemsSolutions
  • 1Take Fivenot published
  • 2Chain Reactionnot published
  • 3Lego Movienot published
Mystery Lab1 problemThe Future Circular ColliderProblemSolution
  • 4Mystery LabThe Future Circular ColliderProblemSolution

2021

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Speed-time graph of a ball turning around*not published
  • 2Horizontal velocity graph of a ball turning around*not published
  • 3Force balance across two pivoted rods*not published
  • 4Westward drift when jumping to different heights*not published
  • 5Shortest time between two cities with bounded acceleration*not published
  • 6Work to lift a submerged bucket to the surface*not published
  • 7Speed change from a slowly moving wall*not published
  • 8Coins to sample for a precise total weight*not published
  • 9Flight time to simulate Mars gravity*not published
  • 10Final spin rate after a stone slides to a disk's rim*not published
  • 11Ranking fall times of three projectiles*not published
  • 12Maximum launch speed before a spring platform slips*not published
  • 13Acceleration directions on a semicircular racetrack*not published
  • 14Scaling of swim speed with a bacterium's size*not published
  • 15Ramp angle that minimizes descent time*not published
  • 16True statements from a velocity-position graph*not published
  • 17Comparing accelerations of points on clock hands*not published
  • 18Shape of a smoke plume after a wind gust*not published
  • 19Gravitational field near a cavity in a planet*not published
  • 20Speed of the Moon's day-night terminator*not published
  • 21Ramp angle maximizing a rolling sphere's kinetic energy*not published
  • 22Angle condition for an icebreaker to submerge ice*not published
  • 23Period of oscillation on a nonlinear spring*not published
  • 24Speed reduction for two satellites to meet*not published
  • 25Fraction of ejected pellets that land back on a planet*not published
USAPhO
Problems6 problemsProblemsSolutions
  • 1A1. Toffee Puddingnot published
  • 2A2. Flashlightnot published
  • 3A3. Electroneeringnot published
  • 4B1. Disk Jockeynot published
  • 5B2. Hot Pocketnot published
  • 6B3. The Mad Hatternot published
USAPhO Plus
Problems3 problemsProblemsSolutions
  • 1The Jet Streamnot published
  • 2The Dark Forestnot published
  • 3Great Hallnot published

2020

USA

F=ma
Exam A25 problemsProblemsSolutions
  • 1Initial speed of a ball that bounces and loses energy*not published
  • 2Fastest point on a ball rolling in a right-angle chute*not published
  • 3Scaling of an axe's burial depth with kinetic energy*not published
  • 4Joint force in a spinning square of rods*not published
  • 5Best shaking frequency for a horizontally shaken pendulum*not published
  • 6New orbit radius after a star loses mass*not published
  • 7Which thrown objects return to the space station*not published
  • 8Acceleration-position graph from a velocity-position graph*not published
  • 9Tension at the end of a pulley string system*not published
  • 10Height a block is raised in a pulley system*not published
  • 11Maximum mass hung from a steel rod by tensile strength*not published
  • 12Locating a mass in a rod from two pivot periods*not published
  • 13Angular momentum after a ballerina releases a pen*not published
  • 14Acceleration graph for stacked blocks with static and kinetic friction*not published
  • 15Gauge pressure beneath two liquid layers*not published
  • 16Scaling of jump height from coalescing droplets*not published
  • 17Scale reading while rain accumulates on a hat*not published
  • 18Kinetic energy after a rotating rod's off-center collision*not published
  • 19Speed of a plate atop rolling cylinders*not published
  • 20Average power dissipated driving against fluctuating wind*not published
  • 21Minimum force to tip a many-legged table*not published
  • 22Frame dependence of momentum and energy change in a collision*not published
  • 23Scaling of spring constant uncertainty with applied force*not published
  • 24Vertical offset of a mass swinging on a zero-length spring*not published
  • 25Stability of a ball held by two horizontal springs*not published
Exam B25 problemsProblemsSolutions
  • 26Cycle time for a ball bouncing between floor and ceiling*not published
  • 27Moment of inertia of a square plate about its diagonal*not published
  • 28Angular frequency of a conical pendulum*not published
  • 29Identifying the free-fall segment of a zero-g flight*not published
  • 30Closest approach of a dust mote falling toward the sun*not published
  • 31Safe loading region on a three-legged table*not published
  • 32Matching orbital periods after two thruster burns*not published
  • 33Time for a block to stop slipping on a conveyor belt*not published
  • 34Equilibrium length of a spring-string over a pulley*not published
  • 35Oscillation period of two masses linked by a spring*not published
  • 36Work done by a person walking down an escalator*not published
  • 37Stress distribution in a cantilevered platform*not published
  • 38Period of a disk oscillating about an edge pivot*not published
  • 39Stopping distance of a block colliding with smaller blocks*not published
  • 40Effect of closing a valve on a piston's height*not published
  • 41Drop height for two masses to collide at equilibrium*not published
  • 42Spring compression for a mass to rebound to its start*not published
  • 43Tension ratio in a spring-linked extendable arm*not published
  • 44Energy and angular velocity of a reshaping spinning object*not published
  • 45Scale reading with a floating balloon inside a box*not published
  • 46Correct relation for standing up on a moving swing*not published
  • 47Relative sliding distance of a mass on a frictional block*not published
  • 48Matching drag force by scaling a fish's swim speed*not published
  • 49Maximum speed of the last box in elastic collisions*not published
  • 50Observation time needed for 1% decay rate precision*not published
USAPhO
Problems6 problemsProblemsSolutions
  • 1A1. Braking upnot published
  • 2A2. Swoosh!not published
  • 3A3. Spin Cyclenot published
  • 4B1. String Cheesenot published
  • 5B2. Mirror Mirror on the Wallnot published
  • 6B3. Real Expansionnot published

2019

USA

F=ma
Exam A25 problemsProblemsSolutions
  • 1Net force on a coin approaching terminal velocity*not published
  • 2Comparing kinetic energy in elastic versus inelastic collisions*not published
  • 3Maximum height off a moving curved wedge*not published
  • 4Oscillation period from a suddenly released spring*not published
  • 5Minimum force to roll a cylinder over a step*not published
  • 6Bouncing ball's timing and spacing down an incline*not published
  • 7Rod tension when a swinging mass is horizontal*not published
  • 8Trajectory type from a given orbital speed*not published
  • 9Ground speed of a point on a rolling wheel*not published
  • 10Moment of inertia of a half annulus*not published
  • 11Uncertainty in a car's measured speed*not published
  • 12Time between bounces of a decaying steel ball*not published
  • 13Power scaling for juggling more balls*not published
  • 14Coriolis deflection of a fired bullet*not published
  • 15Horizontal velocity of a tumbling launched rod*not published
  • 16Depth error from ignoring sound travel time*not published
  • 17Minimum force to hold a disc in a spring launcher*not published
  • 18Exit speed of a wedge from a torque launcher*not published
  • 19Angular frequency of a submerged pendulum*not published
  • 20Stability condition for a spring-supported rod*not published
  • 21Comparing orbits inside and at the edge of a dust cloud*not published
  • 22Tension ratio in a two-string conical pendulum*not published
  • 23Spin direction of a sphere pushed on a sliding slab*not published
  • 24Period ratio in an anisotropic 2D potential*not published
  • 25Water jet speed from a hole in a turning car's beaker*not published
Exam B25 problemsProblemsSolutions
  • 26Friction energy lost sliding down then along level ground*not published
  • 27Comparing densities of wood and styrofoam by floating*not published
  • 28Potential energy ratio of two springs in series*not published
  • 29Collision time for two mutually attracting masses*not published
  • 30Gravitational acceleration profile inside a layered Earth*not published
  • 31Density profile of a self-gravitating spinning cylinder*not published
  • 32Wire tension change when a boat floats in a trough*not published
  • 33Scale reading while pressing with both hands*not published
  • 34Force holding a ball between a wall and a board*not published
  • 35Cart speed after a ball settles in its bowl*not published
  • 36Moment of inertia of a square with a circle removed*not published
  • 37Comparing energy and angular momentum of two orbits*not published
  • 38Wetted region on a wall from a faster water jet*not published
  • 39Initial speed for a ball to bounce back to its start*not published
  • 40Estimating air resistance on a swung mass*not published
  • 41Return time of a spinning, sliding hoop*not published
  • 42Comparing return times of a hoop and a disk*not published
  • 43Dominant uncertainty source in a race time estimate*not published
  • 44Acceleration graph of a wide-swinging physical pendulum*not published
  • 45Descent time compared to ascent time on a frictional ramp*not published
  • 46Normal force on a freight car filling with grain*not published
  • 47Minimum pressure to bruise a dropped apple*not published
  • 48Maximum track amplitude for a comfortable train ride*not published
  • 49Acceleration of a rope end pulled over a pulley*not published
  • 50Best measurement choice to minimize uncertainty in g*not published
USAPhO
Problems6 problemsProblemsSolutions
  • 1A1. Collision Coursenot published
  • 2A2. Green Revolutionnot published
  • 3A3. Electric Slidenot published
  • 4B1. Strain in the Membranenot published
  • 5B2. Stellar Black Boxnot published
  • 6B3. Pitfallnot published

2018

USA

F=ma
Exam A25 problemsProblemsSolutions
  • 1Velocity graph of an object thrown up with air drag*not published
  • 2Center of mass speed in an elastic collision*not published
  • 3True statements about an elastic collision, except one*not published
  • 4Best impulse timing to raise a satellite's energy*not published
  • 5Direction an inclined plane accelerates*not published
  • 6Speed of a crate pushed up a ramp*not published
  • 7Shortest time to travel a distance with bounded acceleration*not published
  • 8Speed of a point on a disk rolling inside a hoop*not published
  • 9Normal force on a stick lifted by one end*not published
  • 10Acceleration graph of an object thrown up with air drag*not published
  • 11Oscillation frequency after cutting a spring in half*not published
  • 12Uncertainty in g from a pendulum experiment*not published
  • 13Young's modulus of a tightrope cable*not published
  • 14Period ratio of a triple pendulum in two directions*not published
  • 15Kinetic energy change of a decaying satellite orbit*not published
  • 16Astronaut's jump path on a rotating space station*not published
  • 17Shape of a sand stream from a turning helicopter*not published
  • 18Tension difference at top and bottom of a vertical circle*not published
  • 19Scaling of pressure from falling raindrops*not published
  • 20Potential energy of a spring cut in half*not published
  • 21Time for a spinning ball to roll without slipping*not published
  • 22Water flow rate into a sinking boat*not published
  • 23Rotational acceleration of a ball versus ramp angle*not published
  • 24Period change of a pendulum with increasing energy*not published
  • 25Comparing uncertainty in combined pendulum measurements*not published
Exam B25 problemsProblemsSolutions
  • 26Acceleration graph of falling clay hitting the ground*not published
  • 27Energy dissipated by friction on an incline*not published
  • 28Kinetic energy after a perfectly inelastic collision*not published
  • 29Conserved quantities in a basketball bounce*not published
  • 30Ratio of spin-up to spin-down acceleration for a disk*not published
  • 31Scaling of beam deflection with length*not published
  • 32Best shaking frequency to grow a pendulum's swing*not published
  • 33Rotational velocity graph of a sliding, spinning ball*not published
  • 34Possible outcomes of an elastic ball collision*not published
  • 35Buoyant force on a balloon at increasing depth*not published
  • 36Transverse wave speed on a faster-spinning rope*not published
  • 37Ball's trajectory seen from a rotating space station*not published
  • 38Force to slide a block off a stacked block*not published
  • 39Pull angle for a spool to slip without rotating*not published
  • 40Scale reading while lifting a textbook*not published
  • 41Engine power and angle of attack in a tailwind*not published
  • 42Maximum speed of a pogo stick after landing*not published
  • 43Equivalent spring constant of two springs in parallel*not published
  • 44Uncertainty in a measured speed of sound*not published
  • 45Maximum pull before a string slips off a table*not published
  • 46Pivot position for the fastest bar oscillation*not published
  • 47Motion of the second mass in a pulley pair*not published
  • 48Timing of a rotating two-mass system*not published
  • 49Gravitational pull from a hemispherical shell*not published
  • 50Best way to reduce uncertainty in a wire's surface area*not published
USAPhO
Problems6 problemsProblemsSolutions
  • 1Problem A1not published
  • 2Problem A2not published
  • 3Problem A3not published
  • 4Problem B1not published
  • 5Problem B2not published
  • 6Problem B3not published

2017

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Friction needed for a motorcycle on a vertical wall*not published
  • 2Spring oscillation frequency during free fall*not published
  • 3Shell displacement after a ball settles inside it*not published
  • 4Evolving pattern of accelerating cars in traffic*not published
  • 5Optimal launch angle off a cliff for maximum range*not published
  • 6Balancing a four-mass mobile*not published
  • 7Kinetic energy growth of a snow-collecting train*not published
  • 8Minimum engine power for a snow-collecting train*not published
  • 9Ranking base forces from water in different flasks*not published
  • 10Gauge pressure at the bottom of a milk jug*not published
  • 11Impact point for a struck rod not to rotate*not published
  • 12Impact mass for a struck rod not to rotate*not published
  • 13Pulley tension as one mass grows without bound*not published
  • 14Which rolling shape accelerates fastest downhill*not published
  • 15Comparing rolling sphere speeds after a fixed drop*not published
  • 16Speed of a rod's end against a slanted wall*not published
  • 17Initial speed of an object falling the last 60 meters*not published
  • 18Point of zero acceleration on a rolling disk*not published
  • 19Friction coefficient from a puck's speed-time graph*not published
  • 20Maximizing momentum transfer in an inelastic collision*not published
  • 21Maximum momentum transfer in an elastic collision*not published
  • 22Maximizing energy transfer in an elastic collision*not published
  • 23Wave pulse travel time on a stretched spring*not published
  • 24Maximum spring compression in a free-space collision*not published
  • 25Orbital speed at perigee versus a nearby point*not published
USAPhO
Problems6 problemsProblemsSolutions
  • 1Problem A1not published
  • 2Problem A2not published
  • 3Problem A3not published
  • 4Problem A4not published
  • 5Problem B1not published
  • 6Problem B2not published

2016

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Angular velocity ratio of a car's front wheels in a turn*not published
  • 2Cylinder floating fraction after adding oil*not published
  • 3Retarding force of snow catching a falling book*not published
  • 4Acceleration of a bead sliding down a helix*not published
  • 5Possible path of a particle in an accelerating box*not published
  • 6Returning path of a particle in an accelerating box*not published
  • 7Period of a mass on a symmetric stretched spring*not published
  • 8Kepler's laws under a modified force law*not published
  • 9Speed of a bead leaving a frictionless sphere*not published
  • 10Accelerations after cutting the upper string*not published
  • 11Scaling of maximum speed with a car's mass*not published
  • 12Floating block in an upward-accelerating container*not published
  • 13Bounding the speed ratio of the struck object*not published
  • 14New frequency after moving a spring's attachment point*not published
  • 15New frequency after moving a spring's fixed post*not published
  • 16Landing distance of a ball rolled off a truck*not published
  • 17Critical velocity for a spinning ball to stop*not published
  • 18Total time for a spin-up, coast, spin-down motion*not published
  • 19Graph relating a bead's launch height and landing distance*not published
  • 20Friction needed for a prism to topple, not slide*not published
  • 21Horizontal displacement of a falling rod-mass system*not published
  • 22Speed of a falling rod-mass system before impact*not published
  • 23New radius of a spinning elastic ring*not published
  • 24Moment of inertia of a hexagon from a triangle*not published
  • 25Comparing measurement uncertainties for a rod's length*not published
USAPhO
Problems6 problemsProblemsSolutions
  • 1Problem A1not published
  • 2Problem A2not published
  • 3Problem A3not published
  • 4Problem A4not published
  • 5Problem B1not published
  • 6Problem B2not published

2015

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Crossing time for a boat in a flowing river*not published
  • 2Average speed over a two-segment car trip*not published
  • 3New airplane speed after doubling engine power*not published
  • 4Acceleration versus applied force with static friction*not published
  • 5Miss distance of a projectile hitting a cliff*not published
  • 6Center of mass speed after cart collisions*not published
  • 7Final speed of a cart chain in inelastic collisions*not published
  • 8Final speed of a cart chain in elastic collisions*not published
  • 9Final speed of a ball after an oblique collision*not published
  • 10Kinetic energy change in an oblique collision*not published
  • 11Fraction of a sphere submerged in oil*not published
  • 12True statement about a pendulum's acceleration*not published
  • 13Direction of acceleration for a swinging pendulum*not published
  • 14Tension in a rope on a spinning rod*not published
  • 15Work done pulling masses toward a spinning rod's center*not published
  • 16True statements from a potential energy graph*not published
  • 17Maximum energy stored in a spinning flywheel*not published
  • 18Functional relationship from a log-log plot*not published
  • 19Oscillation frequency of a U-tube manometer*not published
  • 20Height difference in an oil-water manometer*not published
  • 21Total bounce time of a ball with restitution*not published
  • 22Kinetic energy at the bottom of inclines with friction*not published
  • 23Maximum spring displacement after a falling collision*not published
  • 24Frequency ratio of two tensioned strings*not published
  • 25Frequency ratio of coupled cart oscillations*not published
USAPhO
Problems6 problemsProblemsSolutions
  • 1Problem A1not published
  • 2Problem A2not published
  • 3Problem A3not published
  • 4Problem A4not published
  • 5Problem B1not published
  • 6Problem B2not published

2014

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Angular momentum and acceleration directions in a turn*not published
  • 2Force direction a rolling ball exerts on a ramp*not published
  • 3Volume of an object pressed fully underwater*not published
  • 4Particle counts for exploding fragments*not published
  • 5Lean angle of a unicyclist in a turn*not published
  • 6Bouncing block inside a frictionless box*not published
  • 7Mass of a balanced meter stick*not published
  • 8Oscillation period after cutting a spring in half*not published
  • 9Velocity from a force-time graph*not published
  • 10Time before a spinning string breaks*not published
  • 11Kinetic versus potential energy of a spring oscillator*not published
  • 12Scaling of a paper helicopter's fall time with height*not published
  • 13Scaling of a paper helicopter's fall time with rotor radius*not published
  • 14Fraction of kinetic energy that is translational for a rolling disk*not published
  • 15Maximum speed of a constant-power car on an incline*not published
  • 16Bounding the speed ratio in a 1D collision*not published
  • 17Surface gravity of an expanding dust cloud*not published
  • 18Scale readings in a pulley-box puzzle*not published
  • 19Shape of a cable trailing a moving helicopter*not published
  • 20Rocket firing time to spin up a space station*not published
  • 21Comparing accelerations for two different pulleys*not published
  • 22Angular momentum after shifting mass between orbiting bodies*not published
  • 23Impulse from firing a bowling ball launcher*not published
  • 24Maximum turn angle from firing a launcher*not published
  • 25Rebound height on a frictional second ramp*not published
USAPhO
Problems6 problemsProblemsSolutions
  • 1Problem A1not published
  • 2Problem A2not published
  • 3Problem A3not published
  • 4Problem A4not published
  • 5Problem B1not published
  • 6Problem B2not published

2013

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Time for train cars to pass a fixed point*not published
  • 2Throw speed for a ball to bounce off a wall*not published
  • 3Time of flight statements for a fixed range throw*not published
  • 4Maximum angle before a two-legged sign collapses*not published
  • 5Maximum downward velocity from a scale reading*not published
  • 6Building height from an elevator's scale reading*not published
  • 7Comparing kinetic energies at equal momentum*not published
  • 8Kinetic energy of a braking truck*not published
  • 9Momentum of a braking truck*not published
  • 10Telling a solid ball from a hollow sphere*not published
  • 11Normal force from cubes sliding off a wedge*not published
  • 12Rolling acceleration of a fluid-filled shell*not published
  • 13Is Saturn's ring part of the planet or its moons*not published
  • 14Center of mass velocity before an elastic collision*not published
  • 15Portion of a floating rod above water*not published
  • 16Collapse time of a self-gravitating cloud*not published
  • 17Tension in a rod orbiting a planet*not published
  • 18Identifying a valid final state after internal forces*not published
  • 19Angle of maximum tension in a swinging pendulum*not published
  • 20Maximum tension in a swinging pendulum*not published
  • 21Period scaling with a pendulum's length*not published
  • 22Tension in the Achilles tendon while standing on toe*not published
  • 23Spring constant of a bungee cord*not published
  • 24Maximum extension of a bungee cord*not published
  • 25Horizontal force to push a box up an incline*not published
Semifinal
Problems6 problemsProblemsSolutions
  • 1Problem A1not published
  • 2Problem A2not published
  • 3Problem A3not published
  • 4Problem A4not published
  • 5Problem B1not published
  • 6Problem B2not published

2012

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Height of a falling drop between drips*not published
  • 2Minimum launch angle for height to exceed range*not published
  • 3Angle at which a triangle topples on an incline*not published
  • 4Speed of the third fragment in an explosion*not published
  • 5Kinetic energy lost in a head-on collision*not published
  • 6Time for two vertical cannonballs to collide*not published
  • 7Where two vertical cannonballs collide*not published
  • 8Spring compression of a sliding block with friction*not published
  • 9Escape velocity from surface gravity and radius*not published
  • 10Ranking rolling times down an incline*not published
  • 11Forces on a box pulled by a pulley rope*not published
  • 12String tensions as a hoop rotates*not published
  • 13Maximum speed from a force-position graph*not published
  • 14Force to hold a drilled cylinder at rest*not published
  • 15Maximum constant speed of a constant-power car*not published
  • 16Vibration frequency of a spring block in an accelerating cart*not published
  • 17Period-amplitude relation from a log-log plot*not published
  • 18What stays the same when a spring-mass box falls*not published
  • 19Pipe radius pumping water with a motor*not published
  • 20Scale readings for a partly submerged block*not published
  • 21Height change of a stacked spring system*not published
  • 22Units of sound intensity in SI base units*not published
  • 23Methods that cannot measure local gravity*not published
  • 24Spring constant of a rotating triangle of masses*not published
  • 25Ranking speeds in circular and elliptical orbits*not published
Semifinal
Problems6 problemsProblemsSolutions
  • 1Problem A1not published
  • 2Problem A2not published
  • 3Problem A3not published
  • 4Problem A4not published
  • 5Problem B1not published
  • 6Problem B2not published

2011

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Distance from an average speed with a stop*not published
  • 2Ranking average accelerations from velocity graphs*not published
  • 3Ranking maximum velocities from velocity graphs*not published
  • 4Ranking distances traveled from velocity graphs*not published
  • 5Estimating Earth's orbital acceleration*not published
  • 6Inelastic collision of sliding children*not published
  • 7Kinetic energy change of a spinning skater*not published
  • 8Visible fraction of a floating block*not published
  • 9Length of a spring pulled from both ends*not published
  • 10What increases a pendulum's period*not published
  • 11Water level as a cup sinks*not published
  • 12Compressive force in a triangle of gravitating masses*not published
  • 13Critical pull angle on a rolling spool*not published
  • 14Spacing weights for a uniform beat*not published
  • 15Spring constant from an oscillator's speed*not published
  • 16Force to hold a rope against a railing*not published
  • 17Speed of a box after a rope slips*not published
  • 18Rebound height on a second frictional ramp*not published
  • 19Return time in a 2D restoring force field*not published
  • 20Maximum distance from origin in a force field*not published
  • 21Dimensional analysis of a gas pressure vessel*not published
  • 22Engine RPM for maximum power*not published
  • 23Why a projectile's parabola isn't an ellipse*not published
  • 24Friction power in a turntable bearing*not published
  • 25Matching a rolling cylinder and sliding block's descent*not published
Semifinal
Problems6 problemsProblemsSolutions
  • 1Problem A1not published
  • 2Problem A2not published
  • 3Problem A3not published
  • 4Problem A4not published
  • 5Problem B1not published
  • 6Problem B2not published

2010

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Fastest point on a position-time graph*not published
  • 2Fastest point on a velocity-time graph*not published
  • 3Fastest point on an acceleration-time graph*not published
  • 4Reaction time before a falling piano lands*not published
  • 5Time-of-flight difference for two projectiles*not published
  • 6Height-to-range ratio of a projectile*not published
  • 7Angular momentum at the blackout limit*not published
  • 8Maximum acceleration of a car climbing a hill*not published
  • 9Angle of a hanging mass in an accelerating car*not published
  • 10Acceleration of a sliding two-block stack*not published
  • 11Equilibrium ratio in a three-mass pulley system*not published
  • 12Time for a falling ball to triple its kinetic energy*not published
  • 13Maximum height off a ramp for a rolling ball*not published
  • 14Sliding distance after an elastic collision*not published
  • 15Height reached sliding onto a movable ramp*not published
  • 16Final speed sliding off a movable ramp*not published
  • 17Gravitational potential energy of a tetrahedron of masses*not published
  • 18Force from a potential energy graph*not published
  • 19Possible motion in a given potential*not published
  • 20Total energy of a particle in a potential well*not published
  • 21Scaling of a ball's gravitational self-energy*not published
  • 22Direction a balloon leans in a turning car*not published
  • 23Balancing forces in a branching water flow*not published
  • 24Moment of inertia of a disk with a hole*not published
  • 25New speed after circularizing an elliptical orbit*not published
Semifinal
Problems6 problemsProblemsSolutions
  • 1Problem A1not published
  • 2Problem A2not published
  • 3Problem A3not published
  • 4Problem A4not published
  • 5Problem B1not published
  • 6Problem B2not published

2009

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Average pressure from a falling apple's impact*not published
  • 2Long-term motion after elastic three-block collisions*not published
  • 3Long-term motion after inelastic three-block collisions*not published
  • 4Force on a spacecraft floor during acceleration*not published
  • 5Comparing angular momenta of orbiting satellites*not published
  • 6Speed at fixed height versus launch angle*not published
  • 7Acceleration of a bird from speed and distance*not published
  • 8Angular acceleration from an angular velocity graph*not published
  • 9Net angle turned from an angular velocity graph*not published
  • 10Distance between two apples thrown up and down*not published
  • 11Work done from an acceleration-position graph*not published
  • 12Work done hoisting a sidekick up a building*not published
  • 13Comparing torques on a balanced seesaw*not published
  • 14Conserved quantities in a bullet-block collision*not published
  • 15Friction coefficient of a dragged suitcase*not published
  • 16Oscillation frequency of two masses on a spring*not published
  • 17Which quantities can be measured in SI units*not published
  • 18Pendulum period with an obstructing peg*not published
  • 19Maximum height of a maximum-range throw*not published
  • 20Final speed after a 2D inelastic collision*not published
  • 21Gravitational potential energy of a binary star*not published
  • 22Orbital period of a binary star*not published
  • 23When spring power delivery is maximum*not published
  • 24Tipping versus slipping of a block on an incline*not published
  • 25Angular momentum of two discs stopping each other*not published
Quarterfinal
Problems4 problemsProblemsSolutions
  • 1Estimating Fomalhaut's mass from its planet*not published
  • 2Vertical toss with linear drag*not published
  • 3Work to insert a slab in a charged capacitor*not published
  • 4Maximizing power to parallel bulbs*not published
Semifinal
Problems6 problemsProblemsSolutions
  • 1Problem A1not published
  • 2Problem A2not published
  • 3Problem A3not published
  • 4Problem A4not published
  • 5Problem B1not published
  • 6Problem B2not published

2008

USA

F=ma
Problems25 problemsProblemsSolutions
  • 1Acceleration from speed and distance*not published
  • 2Diagonal displacement of a cockroach in a cube*not published
  • 3Velocity from a position-time graph*not published
  • 4Maximum displacement from a velocity-time graph*not published
  • 5Acceleration graph from a velocity-time graph*not published
  • 6Range of a projectile at a smaller angle*not published
  • 7Child jumping onto a moving sled*not published
  • 8Friction supporting riders on a spinning wall*not published
  • 9Momentum equations for a 2D collision*not published
  • 10Mass from a force-acceleration dataset*not published
  • 11Friction coefficient from a force-acceleration dataset*not published
  • 12Rotational kinetic energy about an off-center axis*not published
  • 13Oscillation amplitude with a stiffer spring*not published
  • 14Kinetic energy of a reeled-in rotating tether*not published
  • 15Maximum carpenter mass on a tabletop*not published
  • 16Acceleration of a mass on a vertical spring*not published
  • 17Equilibrium shift of a spring in an accelerating box*not published
  • 18Maximum speed falling toward a ring*not published
  • 19Acceleration of a constant-power car*not published
  • 20Deflection of a cantilevered steel beam*not published
  • 21Two-body versus three-body particle decay*not published
  • 22Minimum speed for a bullet-pendulum loop*not published
  • 23Comparing planets of equal density*not published
  • 24Total bounce time of a ball with restitution*not published
  • 25Minimum orbital distance of a mislaunched satellite*not published
Quarterfinal
Problems4 problemsProblemsSolutions
  • 1Particle escaping a charged sphere*not published
  • 2Off-center impulse on a pool ball*not published
  • 3Block sliding in a frictional circular track*not published
  • 4Beads sliding down a vertical hoop*not published
Semifinal
Problems6 problemsProblemsSolutions
  • 1Problem A1not published
  • 2Problem A2not published
  • 3Problem A3not published
  • 4Problem A4not published
  • 5Problem B1not published
  • 6Problem B2not published

2007

USA

F=ma
Problems38 problemsProblemsSolutions
  • 1When a parametric path crosses the x-axis*not published
  • 2Acceleration from a velocity-time graph*not published
  • 3Average velocity from a position function*not published
  • 4Falling distance in the second second*not published
  • 5Friction pulling a crate up a hill*not published
  • 6Displacement from a quadratic velocity*not published
  • 7Energy cost of speeding up a toy car*not published
  • 8Scaling of gravitational potential energy*not published
  • 9Center of mass of a sliding block and wedge*not published
  • 10Matching angular accelerations of two wheels*not published
  • 11Ranking kinetic energies of spinning shapes*not published
  • 12Balancing a rock on a measuring stick*not published
  • 13Momentum graphs of an elastic collision*not published
  • 14Friction direction on an accelerating car*not published
  • 15Tension in a string unwinding a disk*not published
  • 16Baseball bouncing off a basketball*not published
  • 17Tangential and normal acceleration of a projectile*not published
  • 18Friction coefficient of an ice chute*not published
  • 19Extension of a non-Hookean spring*not published
  • 20Wall collisions and energy loss*not published
  • 21Scaling of a sphere's rotational inertia*not published
  • 22Momentum gained by rockets of different mass*not published
  • 23Apparent weight on a spinning planet*not published
  • 24Launch angle against a constant wind*not published
  • 25Oscillation period of a floating box*not published
  • 26Friction coefficient of a sliding sled*not published
  • 27Artificial gravity after astronauts move inward*not published
  • 28Maximum friction before a braking bike tips*not published
  • 29Maximum deceleration before a braking bike tips*not published
  • 30Braking bike with unequal front and rear friction*not published
  • 31Radius of gyration of a uniform rod*not published
  • 32Oscillation frequency of a swinging rod*not published
  • 33Maximum oscillation frequency of a swinging rod*not published
  • 34Angular momentum of a string winding around a post*not published
  • 35Speed of an object winding on a tether*not published
  • 36Length of rope wound around a post*not published
  • 37Elastic collision breaking a tethered cord*not published
  • 38Kinetic energy after an elastic tethered collision*not published
Semifinal
Problems6 problemsProblemsSolutions
  • 1Problem A1not published
  • 2Problem A2not published
  • 3Problem A3not published
  • 4Problem A4not published
  • 5Problem B1not published
  • 6Problem B2not published