Work-Energy Theorem (College Board AP® Physics 1: Algebra-Based): Study Guide

An apple of mass 75 g is thrown vertically upwards with an initial velocity of 18 m/s. According to the work-energy theorem, which of the following is the maximum height the apple can reach assuming air resistance is negligible?

A:

B:

C:

D:

The correct answer is D

Analyze the scenario

• The question mentions the work-energy theorem, so this is the method that should be used

• The force of gravity acts on the apple as the ball is displaced and acts parallel to the displacement

• The force and the displacement are in opposite directions, , therefore:

• As the apple reaches its maximum height, its velocity is zero as it changes direction; therefore, its kinetic energy is also zero

Step 1: List the known quantities

• Mass,

• Initial velocity,

• Final kinetic energy,

• Acceleration due to gravity at Earth's surface,

Step 2: State the work-energy theorem equation in terms of the apple

Step 3: Rearrange the equation to solve for the maximum height,

Step 4: Calculate the maximum height

• This is option D

A block with a mass of is used to compress a spring, with a spring constant of , a distance of . After the block is released, it slides across a horizontal surface which has a coefficient of kinetic friction of . What is the maximum distance the block can reach?

Answer:

Step 1: Sketch out the scenario

Step 2: List the known quantities:

• Mass of block,

• Spring constant,

• Displacement of spring compression,

• Coefficient of kinetic friction,

Step 3: Determine the initial and final positions of the block

• The initial position of the block is taken to be when the spring is compressed to its maximum displacement

• The final position of the block is at its maximum height up the slope

Step 4: State the initial and final kinetic energy of the block

• The block is stationary at its initial position and its final position

• Therefore, both initial and final kinetic energy are zero

Step 5: Consider the work done by the force of the spring on the block

• As the spring is compressed, it gains elastic potential energy

• When the spring is released, the loss of elastic potential energy is equal to the work done by the force applied by the spring on the block

Step 6: Consider the work done on the block on the horizontal surface by the force of kinetic friction

• The energy dissipated by the frictional force is:

• The force of kinetic friction is

• The force acts perpendicular to the displacement but in the opposite direction, so , and , therefore:

• The net work is equal to the work done in compressing the spring plus the work done by friction:

• Therefore, the change in mechanical energy is equal to the work done by the spring

• Rearrange to solve for the distance traveled

• Substitute the known values into the equation to calculate the distance traveled by the block

A problem like this could be solved with kinematic equations, but if the question specifically asks you to use a certain method, you must stick with that.