For several of these problems you'll need to look up some standard moments of inertia.

Part A

Recall the definition of impulse from lecture 14.

Part B

Us the definition of momentum to get the velocity given that you know the mass.

The formula derived in the lecture will help you here.

Use conservation of momentum for Part A. Then check the kinetic energy before and after to see what the change is. You can't assume from the start that this is an elastic collision!

Part A and B: The actual collision is a one dimensional elastic collision where one body is at rest so it obeys the equations we derived for this kind of collision. To find the velocity of the object moving before the collision use conservation of mechanical energy for the motion down the slope.

Part C: Use conservation of energy to find the height the block goes up and convert to distance

For a 2 dimensional elastic collision you can apply 3 equations two of which represent conservation of momentum and one of which corresponds to conservation of energy.

The problem is similar to our billiards example from the lecture. The difference is the masses are not the same, but as they are related to each other you'll find that the actual value of the mass is not important (the ratio however, is)

Use conservation of momentum (initial momentum of the system is zero). Remember that momentum is a vector and should add as such!

Draw a triangle to represent the vector sum of the the two initial momenta (which are equal in magnitude). The third side of this triangle should have length $2mv_{f}$, (from conservation of momentum). Use trigonometry to find the angle you need.

Don't forget to convert your angular velocity from $\mathrm{rpm}$ to $\mathrm{rad/s}$.

To find the radial component of the acceleration you'll need to know the tangential velocity, which you should get by first finding the angular velocity at that time and then converting.

The tangential component of the acceleration can be found from you answer to part A.

As before you need to do some unit conversion. Once you have the angular acceleration you can use one of the rotational motion kinematic equations.

Remember that only force applied tangentially will produce a torque, if you work out this component and multiply that by the radius you'll have your answer.

Part A. Find moment of inertia of the ball and angular acceleration, then use the rotational form of Newton's Second Law to find the torque.

Part B. The torque needs to come from somewhere, but as the distance at which the force is applied is quite different the force will be different too.

One object gains potential energy, another loses it. All **three** gain kinetic energy.

This problem is the opposite of the hoop and disk demo we did in the lecture. There we saw that some of the potential energy needed to be used to turn the hoop, here we will get it back when we go up the incline.

For the time taken, if at any point on the slope the kinetic energy is equally divided in to translational and kinetic energy, what must the translational acceleration be?