AP Physics 1 : Kinetic Energy

Example Questions

Example Question #1 : Kinetic Energy

A bungie jumper of mass  is attached to a bungie with a constant of . The unstretched length of the bungie is . What is the maxmimum velocity of the jumper?

Explanation:

Think about this scenario practically. After the jumper jumps, he will begin accelerating at a rate of . This rate will stay constant until the bungie cord begins to stretch. At this point, the jumper has traveled a distance of . The rate of acceleration will now decrease and ultimately reach a rate of . This is the point at which the force from the bungie cord is equal and opposite to the force of gravity. This is also the point at which the jumper is traveling at his or her maxmium velocity. With all of this in mind, let's start writing expressions for the scenario.

The main expression we will use will be the one for conservation of energy:

Plugging in our expressions for these variables and removing initial kinetic energy, we get:

Rearranging for velocity:

We simply need to find the height distance between the jumper's initial position and the position at which the jumper is traveling at his or her greatest velocity. As previously mentioned, the point of highest velocity is the point at which the force from the bungie cord is equal and opposite to the force of gravity:

Rearranging for , we get:

This is the distance that the bungie is stretched. Therefore, we can say that the total height distance between the initial and final state is the length of the unstretched bungie cord plus the distance the cord has stretched:

Plugging this back into the equation for final velocity, we get:

We have values for all of our variables, so we can simply solve for the final velocity:

Example Question #1 : Kinetic Energy

A few friends are having a water balloon fight. One friend throws a balloon of mass  at another friend on a ledge  above him. If the balloon hits the second friend at a rate of , how much energy did the first friend expend throwing the balloon?

Explanation:

We can use the equation for conservation of energy to solve this problem:

If we set the original height to , we can then remove initital potential energy from the equation. Furthermore, the intital kinetic energy will be the amount of energy that the first friend used to throw the balloon. Substituting in expressions for the final variables, we get:

We know all of the variables, allowing us to solve:

Example Question #1 : Kinetic Energy

A man of mass  is going skydiving. After jumping out of the plane, he travels a distance of  before reaching a terminal velocity of . How much work did air resistance do on the diver before he reached terminal velocity?

Explanation:

We can use the equation for conservation of energy to solve this problem:

Here,  is the work done by air resistance.

If we say that the height at which the diver reaches terminal velocity is equal to 0, we can rewrite:

Plugging in expressions for potential and kinetic energy and rearranging for air resistance, we get:

We have values for each variable, allowing us to solve:

Example Question #2 : Kinetic Energy

Consider the following system:

There is no friction between the block and plane, and the angle measures . If the block starts from rest, what is the velocity of the block after it travels down  of the slope?

Explanation:

We can use the equation for conservation of energy to solve this problem:

If we designate the final height to be zero, we can eliminate final potential energy and initial kinetic energy, since the block starts at rest. Therefore, we get:

Substitute expressions for each type of energy:

Eliminate mass and rearrange for final velocity:

We don't know what the initial height is at this point. We do know what the distance traveled down the slope and angle are, so we can figure out the distance traveled using the following equation:

Rearranging for initial height, we get:

Now we have all of the information we need to solve for the final velocity:

Example Question #5 : Kinetic Energy

A bullet with a mass of  is flying through the air at a velocity of . What is its kinetic energy?

Explanation:

The equation for kinetic energy is:

Plug in known values and solve.

Example Question #6 : Kinetic Energy

A dumbell with a mass of  has been dropped from a height of . What is the amount of kinetic energy in the dumbell after three seconds?

Explanation:

Since the stone starts at rest (it has an initial velocity of zero), its velocity at any time is given as:

At a time of three seconds, the velocity towards the ground is thus:

It's kinetic energy at three seconds is therefore:

Example Question #7 : Kinetic Energy

Toby throws a  baseball straight down from the top of a  ridge at a speed of .

What will the ball's kinetic energy be after one second?

Explanation:

This problem can be approached in multiple ways; two will be shown here:

One method is to focus solely on the kinematics and related equations; since there is an initial velocity of , the velocity at any time (provided the ball hasn't hit the ground) is given by the function:

With this, the kinetic energy can be found:

Another although longer method is to use an energy balance:

Wherein due to conservation of energy, the total energy of state 1 is equivalent to the total energy of state 2.

Now to find  of the ball, determine its height at a time of one second using the kinematic equation for position:

Therefore:

Which confirms the answer we used with the earlier method.

Example Question #1 : Kinetic Energy

An  object travelling at  has how much kinetic energy?

Explanation:

In this question, we are provided with the mass of an object as well as its velocity, and we are being asked to determine its kinetic energy. To do so, we'll need to use the following equation:

Plugging in the values given, we obtain:

Example Question #1 : Kinetic Energy

Two objects, one of mass, , and the other of mass, , are traveling at constant velocity along a frictionless surface. The lighter object travels at , while the heavier travels at . An opposing horizontal force acts upon both objects equally, and brings them to rest. Which object takes longer to slow down, and why?

The heavier object, because the mass is squared in the Kinetic Energy formula

The lighter object, because the mass is square rooted in the Kinetic Energy formula

The objects stop in the same amount of time, as their masses and velocities cancel out.

The lighter object, because the velocity is squared in the Kinetic Energy formula

The heavier object, because the velocity is square rooted in the Kinetic Energy formula

The lighter object, because the velocity is squared in the Kinetic Energy formula

Explanation:

Examining the formula for kinetic energy will allow us to compare the two objects. The formula, , shows us that velocity has a higher influence in the calculation of Kinetic Energy than mass does. The velocity of the lighter object is double that of the heavier object, and because of the velocity's larger influence on the formula, the lighter one takes longer to stop, even though its mass is half that of the heavier object.

Example Question #1 : Kinetic Energy

In an investigation into energy, a student compresses a spring a distance of  from its neutral length. She then puts a cart of mass  against the end of the spring and releases both the spring and the cart. She measures the kinetic energy of the cart after the spring has returned to its neutral length and finds it to be . She then repeats the experiment, but this time using a cart whose mass is  before releasing the spring and the cart. What would the cart's kinetic energy be in the second experiment?