Kinetic Energy - AP Physics 1

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:

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:

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:

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:

Kinetic energy is equal to . The object of mass and velocity therefore has kinetic energy equal to . Let's let the object of mass have velocity . Therefore, its kinetic energy is . We want to find such that the two objects to have the same kinetic energy, so we can equate their two kinetic energies.

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.

The equation for kinetic energy is:

Plug in known values and solve.

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:

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.

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:

The potential energy stored in a spring is given by

So the energy increases with the square of the compression. When the potential energy is converted to kinetic energy of the cart, it increases by a factor of .

The potential energy stored in a spring is given by

So doubling the compression increases the potential energy by a factor of . However, the kinetic energy:

, increases with the square of the velocity, so the velocity only needs to double to increase the cart's kinetic energy by a factor of .

Kinetic energy is given by:

, so the energy is linearly dependent on the mass, but increases with the square of the velocity. So although the mass has doubled, the velocity only needs to decrease by a factor of to maintain the same kinetic energy. In algebraic form:

Simplify.

All of the potential energy stored in the spring is converted to kinetic energy of the cart in both experiments. Although the cart's velocity will be less in the second experiment, its kinetic energy will be the same.

We need to know the final speed of the ball since we have all but this quantity in our formula for kinetic energy,

To find the final velocity we first find the acceleration:

We can find the final velocity by multiplying the acceleration by the time. We then plug this into the formula for kinetic energy and solve. Do not forget to convert grams to kilograms.

The initial velocity of the cork was zero.

First, let's remember what it means to have mole of an atom given its atomic mass unit in Daltons. mole of oxygen with atomic mass of is going to be grams. In kilograms, this would be:

Plugging this into the equation for kinetic energy

, where is the mass, is its velocity.

For this problem, and

Therefore,

The equation for kinetic energy is:

Where is the mass of an object and is the velocity of the object.

We see from the equation that mass has a linear relationship with kinetic energy and the problem statement gives that the vehicles have the same velocity. Because the car has half the mass of the truck and the same speed, the car has half as much kinetic energy as the truck.

Without specific numbers given, we can always substitute easy numbers into the equation to help understand the concept. If we say the velocity in both cases is , and the car's mass is then we have:

We know the truck's mass is double the car's, which yields:

This makes it clear that the car has half the kinetic energy of the truck.

Just before impact, the block has maximum kinetic energy. Once the kinetic energy is determined, we can find the potential energy and therefore the height of the drop. We can use the following formula to determine the kinetic energy:

All of this energy was previously potential energy from which we can determine the height, .