Kinetic energy is given by. We will begin by calculating the car's initial kinetic energy, in terms of the unknown mass of the car : 

.

Next, we will calculate the final energy of the car, also in terms of the unknown mass of the car:

.

To find the ratio of the final to initial kinetic energy, we divide  by . We see that this reduces to  with the both the mass  and  terms cancelling. . Thus, the new kinetic energy is .

So in order to solve this problem, we first should think about where all the energy in the system is coming from and going. Specifically when you pull the spring back, the work you do is turned into potential energy stored in the spring. So:

and the energy transforms into kinetic when you let go, but some of that energy is lost to friction so

    and 

So use the equation below, enter in your known quantities and then solve for the work done by friction.

Plug in known values and solve.

This is an example of conservation of energy. The car starts at the top of the ramp, at height . It has no velocity at this time since it is starting from a rest. Therefore its total energy is  where  is the mass of the car and  is the value of gravitational acceleration.

At the bottom of the loop, all of the potential energy will have been converted into kinetic energy.

As the car traverses the loop and rises above the ground, kinetic energy will be converted back into potential energy. The shape of the loop does not matter in this case -- only the vertical distance between the ground and the car.

In the tallest possible loop, all kinetic energy at the bottom is converted to potential energy at the top. This is the maximum height the car can reach -- there is no additional energy left to continue climbing a taller loop. Therefore, the potential energy at the top of the tallest loop we can build is equal to the kinetic energy at the bottom of the loop. But we have already noted that the kinetic energy at the bottom of the loop is equal to the potential energy at the top of the ramp.

Therefore, we set . We see that  and  cancel, and we are left with  . In other words, the tallest loop you can build is equal to the height of whatever ramp you select. In this example, the tallest loop we can build is . We do not need to know the specific values of  or .

Power is the rate of energy transfer. To raise a  object , a total of  or ( is required. To find the power in Watts (), we divide the total energy required by the time over which the energy must be transferred:

We know that:

and we are looking for the maximum height (vertical displacement) this person can obtain, so we aren't concerned with .

We can apply the conservation of energy:

Masses cancel, so 

Solve for :

  (rounded to simplify our calculations)

so let's plug in what we know

. This is our final answer.

Let's first write down the information we are given:

In order to solve this problem we must apply the conservation of energy, which states  since no friction.

This means that as the project reaches its max height energy is converted from Kinetic Energy (energy of motion) to potential gravitational energy (based off of height).

We can subtract  from  to get the  at its max height

=

so we can solve for the height

where 

therefore 

Let's first write down the information we are given:

In order to solve this problem we must apply the conservation of energy, which states  since no friction.

This means that as the project reaches its max height energy is converted from Kinetic energy (energy of motion) to potential gravitational energy (based off of height).

We can subtract  from  to get the    at its max height

=

so we can solve for the mass

where 

therefore 

The formula for gravitational potential energy is:

To find the potential energy lost, we need to find the potential energy of the ball at two heights, then find the difference. 

Note that the energy was not actually lost; rather, it was converted to kinetic energy. 

Gravitational potential energy is given by the equation:

We are given all the information needed to solve for the potential energy.

Plug in known values and solve.

Recall that the units for energy are Joules. Newtons is the unit for force.

The equation for gravitational potential energy is:

We are given all the information needed to answer the question.

Plug in known values and solve.