AP Physics C: Mechanics Lab Investigations12 Suggested Investigations

The 12 suggested AP Physics C: Mechanics lab investigations

1. 1

   Kinematics with Non-Constant Acceleration (Motion Sensor and Integration)

   BI1: Kinematics and Calculus

   Students record the position of an accelerating object (a cart pushed by a rubber band, a ball in free fall with drag, or a cart on a curved track) using a motion sensor that produces position versus time data at high sampling frequency. From the position data, students compute velocity numerically (as the finite difference of position divided by time interval) and acceleration numerically (as the finite difference of velocity divided by time interval), then compare the measured acceleration to the theoretically predicted value. The key insight is that when acceleration is non-constant, the standard kinematics equations do not apply, and numerical integration of the acceleration data to recover velocity and position is the only valid approach. Students plot acceleration versus time, estimate the area under the curve using a Riemann sum or trapezoidal rule, and compare the result to the measured displacement. This investigation makes the definite integral physically concrete before it appears in FRQ contexts. Key concepts are the derivative as the slope of a function, the integral as the area under a curve, numerical differentiation as finite differences, and the propagation of measurement uncertainty through differentiation.

   On the exam: Appears in FRQ parts asking students to predict how an object's velocity function changes when the force function changes, to set up an integral to compute displacement from a given velocity function, or to identify the slope of a linearized graph as a physical quantity.

2. 2

   Variable Force and Work by Integration (Force Probe and Spring)

   BI3: Conservation Laws

   Students use a force probe to measure force as a function of position as a spring is stretched or compressed, producing a force versus displacement graph. They compute the work done by the spring force by calculating the area under the force-displacement curve using a Riemann sum, trapezoidal rule, or graphical integration tool. They then compare the computed work to the elastic potential energy predicted by one half times the spring constant times displacement squared, verifying the work energy theorem for a variable force. A secondary investigation varies the displacement and confirms that work scales as displacement squared. Key concepts are work as the definite integral of force over displacement, the distinction between constant force (W equals F d) and variable force (W equals the integral of F d x), the spring constant as the slope of the linear force versus displacement graph, and elastic potential energy as the stored work. Students also practice the skill of reading physical meaning from the slope and area of an experimental graph, which is a direct analog of FRQ data analysis parts.

   On the exam: Directly supports FRQ parts asking students to compute work done by a variable force by setting up and evaluating a definite integral, or to explain why the formula W equals F d does not apply when force varies with position.

3. 3

   Newton's Second Law with a Velocity-Dependent Drag Force

   BI2: Force and Dynamics

   Students drop an object through a fluid (glycerin, water, or air) and use a motion sensor or video analysis to record velocity versus time. At terminal velocity, drag force equals weight, giving a direct measurement of drag at that speed. Students then model the drag force as proportional to velocity (Stokes drag) or velocity squared (Newton drag) and use Newton's second law in differential equation form (m dv/dt equals mg minus bv) to predict the velocity function. They compare the predicted exponential approach to terminal velocity with the measured data and extract the drag coefficient from the time constant of the exponential. Key concepts are Newton's second law as a first-order differential equation, terminal velocity as the equilibrium state, the exponential solution to a linear differential equation, and the meaning of the time constant as the timescale for approach to equilibrium. This investigation is among the most calculus-rich in the course and prepares students for FRQ parts that provide a differential equation for motion and ask them to identify the terminal velocity or sketch the velocity function.

   On the exam: Appears in FRQ parts providing a differential equation m dv/dt equals mg minus bv and asking students to identify the terminal velocity, describe the shape of v(t), or compute the velocity at a specific time by separation of variables and integration.

4. 4

   Conservation of Energy with a Variable Force (Roller Coaster or Incline)

   BI3: Conservation Laws

   Students release a cart from rest at a measured height on a curved track and measure its speed at the bottom using a photogate or motion sensor. They compute the initial gravitational potential energy and the final kinetic energy, compare the two to test conservation of mechanical energy, and identify energy loss due to friction by measuring the difference. In a second phase, students apply a known variable braking force (a magnet near the track, or a string pulling at an angle) and compute the work done by that force by integrating the measured force profile, then verify that the total energy change equals the total work done by all forces. Key concepts are the work energy theorem with both conservative and non conservative forces, the energy bar chart as a modeling tool, the conditions under which mechanical energy is conserved, and the identification of energy losses as work done by friction or other non conservative forces. This investigation addresses the most common Unit 3 FRQ error, which is applying conservation of energy when non conservative forces are present.

   On the exam: Directly supports FRQ energy parts that include a friction force, asking students to write the correct energy equation including the work done by friction and to compute the final speed from the corrected energy balance.

5. 5

   Impulse, Momentum, and the Force-Time Integral (Force Plate Collision)

   BI3: Conservation Laws

   Students use a force plate (or force probe mounted on a wall) and a motion sensor to record the force versus time profile of a collision (a cart striking a bumper) and the velocities before and after impact. They compute the impulse as the area under the force versus time curve using graphical integration, then verify that the impulse equals the measured change in momentum. A second phase compares soft versus hard bumpers: the same momentum change is delivered over a longer or shorter collision time, producing a smaller or larger peak force for the same impulse. Key concepts are impulse as the definite integral of force over time, the impulse momentum theorem as the fundamental relationship between impulse and momentum change, the area under a force time graph as a physically meaningful quantity, and the inverse relationship between contact time and peak force at constant impulse. The graphical integration skill transfers directly to FRQ data analysis parts that present a force versus time graph and ask students to compute the impulse or identify the change in velocity.

   On the exam: Supports FRQ parts providing a force versus time graph for a collision and asking students to compute the impulse from the area, find the post-collision velocity from the impulse momentum theorem, or predict how a longer collision time changes the peak force.

6. 6

   Conservation of Momentum in Two Dimensions (Projectile Collisions)

   BI3: Conservation Laws

   Students fire a projectile (a ball launched from a spring gun) at a stationary ball on a table edge, record the landing positions of both balls using carbon paper on the floor, and verify that the vector sum of momenta after the collision equals the initial momentum before the collision. The experiment requires resolving momentum into x and y components and adding them vectorially, making it one of the few investigations that naturally requires two dimensional vector arithmetic. Students also compute the kinetic energy before and after to classify the collision as elastic or inelastic. Key concepts are conservation of momentum as a vector equation, the independence of horizontal and vertical momentum components, projectile kinematics to relate landing position to launch velocity, and the distinction between elastic and inelastic collisions. The two dimensional nature of the investigation builds the vector manipulation skill that FRQ questions on angular momentum (which is a vector with a specific direction) also require.

   On the exam: Supports FRQ collision parts asking students to apply conservation of momentum in two dimensions, compute post-collision speeds and directions, or determine whether kinetic energy was conserved in a described collision.

7. 7

   Rotational Kinematics and Angular Acceleration (Rotating Platform)

   BI4: Rotation

   Students apply a constant torque to a rotating platform via a string and hanging mass, measure the angular position versus time using a rotary motion sensor, and fit the data to a quadratic function to extract the angular acceleration. They compare the measured angular acceleration to the theoretical prediction from Newton's second law in rotational form (net torque equals I times alpha), computing the theoretical moment of inertia of the platform from its geometry and mass. Students then add a known mass at a measured radius and repeat the experiment, observing that the angular acceleration decreases as the moment of inertia increases. Key concepts are rotational kinematics (angular position, angular velocity, angular acceleration as derivatives), the rotational analog of Newton's second law, moment of inertia as the rotational analog of mass, and how adding mass at a larger radius increases moment of inertia more than adding it at a smaller radius. The investigation makes the I equals m r squared dependence on radius experimentally concrete.

   On the exam: Directly supports FRQ rotation parts asking students to predict how angular acceleration changes when mass is added at a given radius, to apply net torque equals I times alpha, or to compute the kinetic energy of a rotating object.

8. 8

   Moment of Inertia by Integration Verification (Uniform Rod)

   BI4: Rotation

   Students measure the oscillation period of a uniform thin rod suspended at different pivot positions and use the period equation for a physical pendulum (T equals 2 pi times the square root of I divided by M g d, where d is the distance from the pivot to the center of mass) to extract the moment of inertia at each pivot. They compare the extracted moment of inertia to the theoretical value derived by integration: I about the center is m L squared divided by 12, and I about a point at distance d from the center is m L squared divided by 12 plus m d squared by the parallel axis theorem. The experiment builds the intuition that moment of inertia depends on both total mass and how that mass is distributed. Key concepts are the physical pendulum as a system whose period reveals the moment of inertia, the parallel axis theorem, moment of inertia by integration, and the difference between the moment of inertia formula for a rod (derived by integration) and for a point mass (m R squared). This is the investigation that most directly prepares students for FRQ parts asking them to derive a moment of inertia by integration.

   On the exam: Appears in FRQ rotation parts asking students to derive the moment of inertia of a uniform rod about its center or end by integration, or to apply the parallel axis theorem to find the moment of inertia about an off-center pivot.

9. 9

   Conservation of Angular Momentum (Spinning Platform and Mass Redistribution)

   BI4: Rotation

   Students rotate on a freely spinning platform or use a rotating disk with movable masses, and observe that pulling masses inward increases the angular velocity while extending them outward decreases it, consistent with conservation of angular momentum when no net external torque acts. Using a rotary motion sensor to record angular velocity and a meter stick and scale to measure mass and radius, students compute angular momentum before and after redistribution and verify that the product I omega is conserved. A second phase uses a falling mass dropped onto the rim of the spinning platform to model an inelastic rotational collision: angular momentum is conserved but kinetic energy is not. Key concepts are angular momentum as the product of moment of inertia and angular velocity, the no-external-torque condition for conservation, the inverse relationship between moment of inertia and angular velocity at constant angular momentum, and the parallel between linear and angular momentum conservation in collisions. The investigation directly models the FRQ scenario of a mass dropped onto a rotating disk, which is documented in Chief Reader Reports as one of the most frequently examined rotation problems.

   On the exam: Directly maps to FRQ problems presenting a mass dropped onto a rotating disk or a skater pulling in arms, asking students to find the final angular velocity using conservation of angular momentum and to determine whether kinetic energy is conserved.

10. 10

    Rolling Without Slipping (Incline with Cylinders and Spheres)

    BI4: Rotation

    Students release different round objects (solid cylinder, hollow cylinder, solid sphere) from rest at the top of an incline and measure their arrival times at the bottom, then predict the order of arrival using conservation of energy with the rolling constraint. The rolling without slipping constraint (v equals omega times R) links translational and rotational speeds, and the total kinetic energy is the sum of translational and rotational kinetic energies. Objects with less of their mass concentrated at the rim (smaller ratio of moment of inertia to mass times R squared) accelerate faster. Students also examine the friction force required for rolling without slipping: friction does no work (because the contact point has zero velocity instantaneously) but provides the torque that accelerates rotation. Key concepts are the rolling without slipping constraint, total kinetic energy as the sum of translational and rotational parts, the race prediction from energy conservation, and friction as the source of torque (not energy dissipation) in rolling without slipping. This investigation addresses the free body diagram error identified most often in Chief Reader Reports for rolling problems.

    On the exam: Supports FRQ energy parts for rolling objects: asking students to write the total kinetic energy at the bottom of an incline, to predict which object arrives first and justify using the moment of inertia fraction, or to draw the free body diagram with the correct friction direction and explain why friction does no work.

11. 11

    Simple Harmonic Motion: Differential Equation Verification (Spring Mass System)

    BI5: Oscillations and Fields

    Students attach a mass to a spring and record position versus time using a motion sensor, then fit the data to a sinusoidal function x(t) equals A cosine (omega t plus phi) and extract amplitude, angular frequency, and phase. They verify that the measured omega equals the square root of k divided by m by independently measuring k (from the slope of a force versus extension graph) and m, and compare the measured period to the theoretical prediction T equals 2 pi divided by omega. In a second phase, students differentiate the position data numerically to compute velocity and acceleration, verify that the acceleration is proportional to negative position (the defining characteristic of SHM), and confirm that the proportionality constant is omega squared. This investigation makes the differential equation d squared x divided by d t squared equals negative omega squared x experimentally concrete: the measured acceleration versus position data should be a straight line through the origin with slope negative omega squared. Key concepts are the SHM differential equation and its sinusoidal solution, the relationship between angular frequency and the physical parameters (spring constant and mass), and the phase relationship between position, velocity, and acceleration in SHM.

    On the exam: Directly supports FRQ SHM parts asking students to write the differential equation for a spring mass system, identify omega from the physical parameters, write the general solution, or predict how the period changes when mass or spring constant is altered.

12. 12

    Gravitational Force and Orbital Motion Verification (Video Analysis or Simulation)

    BI5: Oscillations and Fields

    Students use video analysis software (such as Tracker) applied to a recorded planetary or satellite orbit, or use a computer simulation of a mass orbiting a central force, to measure orbital radius and period for multiple orbit configurations and verify Kepler's third law (T squared is proportional to R cubed). They derive the proportionality constant (4 pi squared divided by G M) from the measured T squared versus R cubed graph and compare it to the theoretical value computed from the known central mass. In a second phase, students verify that the gravitational force at the orbital radius equals the centripetal force required for circular orbital motion, testing Newton's law of universal gravitation in the orbital context. Key concepts are Newton's law of universal gravitation as an inverse square force, the derivation of orbital speed from setting centripetal force equal to gravitational force, Kepler's third law as a consequence of this force law, and gravitational potential energy as the integral of the gravitational force (yielding U equals negative G M m divided by r). The investigation grounds the abstract concepts of gravitational potential and escape velocity in a measurable orbital relationship.

    On the exam: Supports FRQ gravitation parts asking students to derive the orbital speed or period using Newton's second law and the gravitational force, to verify Kepler's third law, or to compute the escape velocity from a planet's surface using energy conservation.

The 12 investigations above are suggested College Board aligned investigations for AP Physics C: Mechanics. They are College Board's guidance, not a fixed required list. The AP Physics C: Mechanics course framework requires that students conduct inquiry based investigations across the 7 course units that collectively develop all 6 Science Practices. Teachers design or select investigations within this framework. The calculus based nature of the course means that data analysis in these investigations often involves numerical differentiation, graphical integration, and fitting experimental data to theoretically derived functional forms, which are skills tested directly in FRQ data analysis parts.