Mathematical Routines (College Board AP® Physics 1: Algebra-Based): Exam Questions

Figure 1

A ball of mass is attached to a rope of length and swings with constant speed in a vertical circle, as shown in Figure 1. At Point A, the ball passes the lowest point in its path, and at Point B, it makes an angle of with the horizontal. The magnitude of the tension in the rope at Point B is equal to three-quarters of the magnitude of the tension in the rope at Point A.

The dots below represent the ball at points A and B. Draw and label the forces (not components) that act on the ball at each point. Each force must be represented by a distinct arrow starting on and pointing away from the appropriate dot. The relative lengths of the arrows should reflect the relative magnitudes of the forces.

Starting with Newton's second law, derive an expression for the tension in the rope as the ball passes Point A. Express your answer in terms of , , , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Determine an expression for the speed of the ball in terms of and physical constants as appropriate.

At Point B, the rope is released, and the ball becomes a projectile with the rope still attached.

Indicate whether the range of the projectile with the rope attached is greater than, equal to, or less than the range of the projectile without the rope attached. Justify your reasoning.

Figure 1

A stunt cyclist builds a ramp that will allow the cyclist to coast down the ramp and jump over several parked cars, as shown above. To test the ramp, the cyclist starts from rest at the top of the ramp, then leaves the ramp, jumps over six cars, and lands on a second ramp.

Figure 1 shows the vertical distance between the top of the first ramp and the launch point, , and the angle of the ramp at the launch point, , as measured from the horizontal. The cyclist travels a horizontal distance of whilst the cyclist and bicycle are in the air. The combined mass of the stunt cyclist and bicycle is .

Figure 2

Figure 2 shows the position, , of the stunt cyclist and bicycle as a function of time for the duration of their vertical motion.

On Figure 3, sketch a graph of the vertical component of the stunt cyclist's velocity as a function of time from immediately after the cyclist leaves the ramp to immediately before the cyclist lands on the second ramp. On the vertical axis, clearly label the initial and final vertical velocity components, and . Take the positive direction to be upwards.

Figure 3

i) Starting with conservation of energy, derive an expression for the speed of the cyclist as they leave the ramp in terms of and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference booklet.

ii) Starting with a kinematic equation, derive an expression for the distance travelled, in terms of , , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference booklet.

In experiment 2, the ramp is modified so that the cyclist now starts from a height of .

Indicate whether the horizontal distance covered by the stunt cyclist in experiment 2, will be less than, equal to, or greater than six cars.

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

Justify your reasoning.

A car of mass travels around a banked curve of radius at a speed slower than the ideal speed of for the given banking angle. The coefficient of static friction between the tires and the road is to be determined.

Draw a free-body diagram for the car on the banked curve.

Starting from Newton's second law, derive an expression for the ideal banking angle required for the car to stay on the curve without friction at the ideal speed.

The car is moving at speed which is slower than the ideal speed.

Derive an expression for the minimum coefficient of static friction required to prevent sliding.

The speed of the car is increased beyond the ideal value.

Indicate whether the direction of the frictional force acts up the incline, down the incline, or perpendicular to it.

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ up the incline, ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ down the incline, ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽perpendicular to the incline.

Justify your reasoning.

Figure 1

A projectile of mass is fired horizontally from a launching device, exiting with a speed , as shown in Figure 1. While the projectile is in the launching device, the impulse given to it is , and the average force exerted on it is . Assume the force becomes zero just as the projectile reaches the end of the launching device.

i) Determine an expression for the time required for the projectile to travel the length of the launching device, in terms of and .

ii) Determine an expression for the mass of the projectile, in terms of and .

Figure 2

The graph in Figure 2 shows the force exerted on the projectile by the launcher as a function of time . The mass of the projectile is .

i) Using the graph in Figure 2, calculate the impulse given to the projectile.

ii) Calculate the launch speed of the projectile.

Figure 3

A block of mass is suspended from the ceiling using a string with negligible mass. The projectile is fired horizontally into the block, as shown in Figure 3. When the projectile strikes the block, the impulse given to the block is. After the collision, the projectile becomes embedded in the block.

If the projectile is much less massive than the block , estimate the approximate magnitude of the impulse given to the block. Justify your estimate using physical principles.

Figure 1

Block 1 of mass is held at rest while an ideal spring of spring constant is compressed by . Block 2 has mass where . At time , Block 1 is released. At time , the spring is no longer compressed and Block 1 immediately collides with and sticks to Block 2. The two-block system moves with constant speed , as shown in Figure 1. Friction between the blocks and the surface is negligible.

The impulse on Block 1 from the spring during the time interval is . The impulse on Block 1 from Block 2 during the collision is .

Indicate whether the magnitude of is greater than, less than, or equal to the magnitude of.

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

Justify your reasoning.

On the axes provided in Figure 2, sketch graphs of the magnitude of the momentum of each block as functions of time from to after . The collision occurs in a negligible amount of time. The grid lines on each graph are drawn to the same scale.

Figure 2

Derive an expression for the velocity of the two-block system after the collision in terms of , , and . Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

In another scenario with the same equipment, Block 1 is used to compress the spring by the same amount as before, and Block 2 is placed in the same initial position. When the spring is released, Block 1 remains attached to the spring as it collides with Block 2.

Indicate whether the velocity of Block 2 immediately after the collision would be greater than, less than, or equal to the velocity of the two-block system in the original scenario. Briefly justify your answer.

Ball 1 has mass and moves horizontally with speed towards a wall. During the collision, Ball 1 is in contact with the wall for time . After the collision, Ball 1 rebounds at half its initial speed.

i) Determine an expression for the impulse exerted by the wall on Ball 1 in terms of and .

ii) Derive an expression for the average force exerted by the wall on Ball 1 in terms of , and . Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

A different ball, Ball 2, moves horizontally towards the wall. Ball 2 has the same mass and initial speed as Ball 1. During the collision, Ball 2 is in contact with the wall for a shorter time than Ball 1. After the collision, Ball 2 rebounds with speed . The average force exerted by the wall is the same in both collisions.

Figure 1

On the dots in Figure 1, draw arrows to represent the impulses exerted on Ball 1 and Ball 2 by the wall during their respective collisions.

• If the impulse is zero, write “zero” next to the dot.

• The impulse, if it is not zero, must be represented by an arrow starting on, and pointing away from, the dot.

• The length of the vectors, if not zero, should reflect the relative magnitude of the impulses exerted on each ball.

Indicate whether the final speed of Ball 2 is greater than, equal to, or less than the final speed of Ball 1.

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

Justify your reasoning using your answer to part b).

Figure 1

In Experiment 1, students release a disk of mass and radius from rest at the top of a ramp that makes an angle with the horizontal. The disk rolls without slipping, as shown in Figure 1. The rotational inertia of the disk is . The frictional force exerted on the disk is .

i) On the diagram in Figure 2, draw and label arrows that represent the forces (not components) that are exerted on the disk as it rolls down the ramp, which is indicated by the dashed line. Each force in your diagram must be represented by a distinct arrow starting on, and pointing away from, the point at which the force is exerted on the disk.

Figure 2

ii) Determine an expression for the net torque exerted on the disk around the center in terms of , , , and physical constants as appropriate.

iii) Determine an expression for the net force exerted on the disk in terms of , , , and physical constants as appropriate.

iv) Derive an expression for the translational acceleration of the center of mass of the disk. Express your answer in terms of , , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 3

In Experiment 1, the disk reaches the bottom of the ramp in time . In Experiment 2, the students release a block of ice from rest from the same height on a ramp inclined at the same angle , as shown in Figure 3. Friction between the block and the ramp is negligible. The block reaches the bottom of the ramp in time .

Indicate whether is greater than, less than, or equal to .

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

Justify your reasoning.

Figure 1

A spring of unknown spring constant is attached to a ceiling. A lightweight hanger is attached to the lower end of the spring, and a motion detector is placed on the floor facing upward directly under the hanger, as shown in the figure above. The bottom of the hanger is 1.00 m above the motion detector.

An object of mass is then placed on the hanger and allowed to come to rest at the equilibrium position, such that the bottom of the hanger is distance below its initial position. The spring is then stretched downward from equilibrium and released at time = 0 s. The motion detector records the height of the bottom of the hanger as a function of time. The output from the motion detector is shown in Figure 2.

Figure 2

i) On Figure 3 below, sketch a free body diagram of the object at equilibrium.

Figure 3

ii) Determine an expression for in terms of , and physical constants as appropriate.

iii) Starting with the equation for the period of a mass on a spring, derive an expression for the frequency of the object on the spring in terms of and physical constants as appropriate.

At time 0.75 s, the system has total kinetic energy and total potential energy ​. At time 1.13 s, the system has kinetic energy ​ and potential energy ​.

i) Indicate whether is greater than, less than, or equal to . Justify your claim using features from Figure 2.

ii) The experiment is repeated. The mass is displaced from equilibrium by the same amount but a new spring is used with spring constant . Predict how Figure 2 will change. Justify your claim.

Figure 1

A pendulum is displaced to angle before being released. Upon being released, the pendulum has angular acceleration . The string has length and the bob at the end has mass .

i) On the diagram in Figure 1, draw and label arrows that represent the forces (not components) that are exerted on the pendulum bob. Each force in your diagram must be represented by a distinct arrow starting on, and pointing away from, the point at which the force is exerted on the bob.

ii) Determine an expression for the tangential acceleration of the pendulum the moment it is released in terms of and physical constants as appropriate.

iii) Derive an expression for the period of this pendulum. Express your answer in terms of , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

In Experiment 1, the pendulum is displaced to before being released. In Experiment 2, the pendulum is displaced to , where and .

Indicate whether the maximum speed of the pendulum in Experiment 1 is greater than, less than, or equal to the maximum speed of the pendulum in Experiment 2.

⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ‎ ‎ ‎‎ ‎ ‎ ‎ ‎ ‎ ‎ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽

Justify your reasoning.

A small block of mass is attached to an ideal horizontal spring of spring constant and is set into simple harmonic motion. The system oscillates on a frictionless surface. The block is displaced by a distance and released from rest. While the block is oscillating, it has a maximum speed . The motion can be described by the equation:

where is the frequency of oscillation.

i) On the axes provided in Figure 1, sketch graphs of position and velocity as a function of time for two complete cycles of motion.

Figure 1

ii) Derive an expression for the magnitude of the maximum acceleration of the block. Express your answer in terms of , , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

iii) Derive an expression for the magnitude of the maximum velocity of the block. Express your answer in terms of , , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

A student places a second identical block onto the first block when it is at maximum displacement. The new system continues oscillating. Air resistance is negligible.

Indicate whether the frequency of oscillation increases, decreases, or remains constant after the second block is added. Justify your reasoning.