1) If you drop both a large and a small ball with the large ball beneath the small, then the small ball can bounce higher into the air than if it was to bounce off the floor directly. If you have a large and small ball handy you could try it! Assume the mass of the large ball is 0.1 kg and the mass of the small ball is 0.01 kg. The radius of the large ball is 30 cm and the radius of the small ball is 10 cm. The balls are initially in contact, with the large ball at a height of 1 m (center of the ball). If the balls are released and fall under gravity to the floor, how high will the small ball bounce up in the air? Assume all collisions to be elastic.

i. How far do the balls initially fall? How fast are they traveling when the large ball hits the ground?

ii. The large ball can be assumed to hit the floor first and rebound back up. At this point, the large ball  is traveling up and the small ball is traveling down. Both have the same magnitude of velocity found in i, however, as the collision was "elastic". Use conservation of momentum in an elastic collision to find  the new velocities of the large and small balls after they collide.

iii. Now the small ball is traveling back up. Given its velocity, how high will it reach?

2) To protect their young in the nest, peregrine falcons will fly into birds of prey (such as ravens) at high speed. In one such episode, a 600 g falcon flying at 20 m/s hit a 1.5 kg raven flying at 9 m/s. The falcon hit the raven at right angles to its original path and bounced back at 5 m/s. (These figures are from a research paper and estimated by the author as he watched this attack occur in New Mexico).

Find the change in angle of the ravens motion and its final velocity after the collision. Was energy conserved in the collision?

This is a simple case of momentum before the collision equals momentum after the collision.

i. Write an equation for the momentum in the direction of the raven as he flies towards the collision.  Put the momentum before the collision equal to the momentum after the collision in this direction.

ii. Do the same analysis for the direction that the falcon was originally flying.

iii. You should have two equations which yield the ravens velocity in both directions after the collision.

Use this to find the magnitude and direction.

iv. Was energy conserved? Calculate the energy before collision, knowing the falcon and ravens initial  velocities and masses. Calculate the energy after the collision, knowing the falcon and ravens velocities after the collision.

3) A wheel starts from rest and rotates with constant angular acceleration to reach an angular speed of 12 rad/s in 3 s. Find the magnitude of the angular acceleration of the wheel and the angle in radians through which it rotates in this time.

Use the linear equations of motion but for angular motion and this should be easy