To solve this problem, we must look at the total energy of the system at two different times: at the beginning before the motion starts, and at the end of the incline.  From there, we can use the conservation of energy to solve for the speed of the object at the bottom of the incline.

Kinetic energy  is related to velocity  and mass  by

Object A's kinetic energy has decreased by . We know that its mass hasn't changed. 

Relate velocity to kinetic energy, while mass is not changing

Since  has changed by a factor of ,  will change by a factor of

Conservation of energy is a very powerful tool for solving some mechanics problems. If we used Newtonian mechanics this would be a lot more difficult. The conservation of energy for the cart denotes:

Where  is the kinetic energy of the system and  is the potential energy.   and  are defined by:

When the cart is at the top of the track it is at rest. This means that there is no initial kinetic energy. When the cart is at the bottom of the track it's height is zero, so it has no final potential energy. Therefore,

Notice that the mass cancels. Solve for :

Convert to miles per hour:

We will use conservation of energy to help us.

We will treat our initial situation as the moment the ball left the hand, and the final situation as the ball at the maximum height.

Mass cancels out

will be zero at our maximum height

Rearranging our equatin to solve for

We then plug in our values

Use conservation of energy:

At the moment of dropping, there will be no velocity and thus no kinetic energy. Right before hitting the ground, there will be no height, and this no potential energy.

Solve for velocity:

Use conservation of energy

At the moment of dropping, there will be no velocity and thus no kinetic energy. Right before hitting the ground, there will be no height, and this no potential energy.

Solve for velocity:

Use definition of momentum:

Use the formula for potential energy due to gravity:

First, use the velocity update kinematic equation:

Then use the definition of kinetic energy:

Since the ramp is frictionless and the disk rolls without slipping, we can infer two things: one conservation of energy applies here, and two we have the condition that , where  is the linear velocity, and  is the angular velocity. For convenience, we will set the potential energy to 0 at the bottom of the ramp, since then for the final energy of the disk will not contain any potential energy terms. Now the initial energy of the disk  will be all potential energy, since the disk starts rolling from rest. Thus , its gravitational potential energy. At the bottom of the ramp, it will not have any potential energy, but it will have two types of kinetic energy: translational kinetic energy due to the fact that the disk itself is moving down the ramp, and rotational kinetic energy due to the fact that the disk is rotating about its center. Thus its final energy  is given by 

Where the first term is the translational kinetic energy contribution, and the second term is the rotational kinetic energy contribution. Now, applying conservation of energy gives us:

Since we are solving for  the linear velocity, we use the fact that  and also we replace the moment of inertia  with  to obtain the following:

Now we solve for  by multiplying each side by  and then taking the square root. We obtain: 

Thus the linear velocity is given by 

The popgun compresses a spring and stores potential energy in the spring. The spring then releases and fires the dart, converting all of the potential energy in the spring into kinetic energy which launches the dart out of the gun at . Therefore, we can use conservation of energy to find the value of the spring constant. The initial energy is given by

Where  is the spring constant and  is the compression distance. Because the dart is sitting in the gun, there is no initial kinetic energy. The final energy is given by 

Where  is the mass of the dart, and  is the velocity of the dart. From the givens in the problem, we know that 

, , and  

Now we apply conservation of energy. We have:

To solve for , we multiply each side of the equation by  to obtain:

Therefore the spring constant is .