Assignment:
Introduction:
Suppose you drop a rubber ball from an initial height ho and watch it strike the floor and bounce back up. If mechanical energy is conserved, then you would expect the ball to bounce back to the same height:
Initial KE + Initial GPE - Final KE + Final GPE - 0 + mgho = 0 + mph1
1) Why does initial kinetic energy equal 0 in the above expression?
2) Why does the final kinetic energy equal 0 in the above expression?
However, you know from experience that even a very good bouncy ball will always bounce a little bit lower than the height from which it was dropped. This means that mechanical energy is not quite conserved.
3) Mechanical energy refers to kinetic energy + potential energy. What are some forms of energy that are not included in mechanical energy?
When the ball strikes the ground, some of the ball's mechanical energy goes into these other forms of energy causing the ball to bounce up a little lower on each bounce. The coefficient of restitution is a measure of how much mechanical energy goes into these other forms of energy.
Coefficient of Restitution:
When two or more bodies collide, the momentum for the system remains constant. This is not necessarily true for the kinetic energy. If the kinetic energy remains constant, we call the collision perfectly elastic. If the kinetic energy goes into deforming the bodies and they stick together, the collision is perfectly inelastic. Clearly, there are NO real perfectly elastic collisions in the macroscopic world. Is the same true on the atomic level?
For a pefectly elastic collision. conservation of kinetic energy and conservation of momentum an be combined to show that the relative velocity of approach (v1,i, - v2,i,) before collision is equal to the relative velocity of separation (v2,f - v1,f) after collision. However, experimentally, in collisions in which some mechanical energy is lost. the above must be replaced by:
(v2,f - v1,f) = e(v1,i - v2,i)
where e is a constant called the coefficient of restitution. The value of e ranges from e = 1 for perfectly elastic collisions to e = 0 for perfectly inelastic collisions.
In our experiment we will be dropping small balls onto a hardened steel slab.
4) Suppose we call the steel slab object 2. What is the slab's velocity before the collision? What is the slab's velocity after the collision?
5) Insert your v2 results into Equation (I) and rewrite the equation below as Equation (2).
6) Suppose you drop the ball from an initial height ho (above the slab). Solve for the velocity of the ball just before it hits the slab in terms of ho. This is velocity vi, in Equation (2). Pay attention to the sign of the velocity!
7) An upward velocity v1,f, the velocity just after impact, will result in the ball rising to a height, h1. Use Solve for v1 in terms of h1.
8) Substitute your expressions for v1,i and v1,f into equation (2) to obtain a new relation for e. Simplify your result and call it Equation (3).
Experiment:
You will use Equation (3) to solve for e, the coefficient of restitution, for a variety of spheres made of different materials. It is recommended that for each sphere, measure at least three different trials dropping the sphere from the same height each time. The spheres are as follows: 5/8" chrome-steel, 5/8" nylon, 5/8" glass, 5/8" wood, 1/2" lead, 5/8" brass, 5/13" aluminum, 5/8" rubber, and 5/16" chrome-steel. Measure the diameters with the micrometer calipers to make sure of each sphere's diameter. Three values of e should be obtained for each sphere. Place these values in a data table, along with values for h0 and h1.
It will help if the spheres are dropped from the center of the tube so they will have less chance of hitting the side as they fall down. Each time a sphere is dropped, the height of its rebound for each sphere should be measured. You may find it helpful to take a slow motion video of the bounce on your cell phone to help you measure the bounce height. The bounce height of the 5/8" chrome-steel ball and the 5/16" chrome-steel ball should be measured especially carefully.
Analysis & Further Investigations:
Q I) Which sphere had the largest coefficient of restitution?
Q2) Is the value of c independent of the initial height ho? Do an experiment to determine the answer.
Q3) Using ideas about kinetic energy transfer during the collision, explain why the smaller chrome-steel sphere rebounds higher than the larger chrome-steel sphere.
Q4) Make a 5/8" sphere of Silly Putty and measure its value of e. Explain why it has a much higher (or much smaller) value of e than the other materials used.
Q5) Suppose we drop a ball and watch it bounce multiple times. Each bounce will be smaller than the last with the bounces eventually becoming too small to even see. Suppose you drop a ball from a height of 1 m and it has a coefficient of restitution of 0.92. How many times will the ball bounce before its bounce height becomes less than 1 mm (too small for you to see)? Hint: Look up the change of base formula for logarithms or use Excel to solve the problem iteratively.