(a) Give the energy transformation equation for one complete ball bounce.

Question

(b) Explain how you could determine the efficiency with which the mechanical energy of a ball bounce is conserved.

Answer 1

Mechanical energy is the sum of kinetic plus all potential energy at a given point and time.  The two main types of potential energy dealt with in Newtonian mechanics are gravitational potential and spring potential.  Since there are no specific springs indicated in this problem, only gravitational potential energy concerns us here (though there is some "springiness" in the ball, which is how a bounce can conserve at least some mechanical energy).

 

Once we know the weight of the object in question, gravitational potential energy depends only on height (above our specified "zero level," which is often the ground).  So, knowing the weight, once we determine an object's height above zero, we can calculate its gravitational potential energy, with just that information.  So, holding an object at some height h, it has gravitational potential energy (GPE) = mgh.  If it is stationary, it has no kinetic energy, since kinetic energy is the energy of motion.

 

If we look at the split-second instant just after the object is released from height h, but before it starts to fall, it has mechanical energy equal to:

 

ME = GPE + KE (by definition) = mgh + 0 (no kinetic energy, because it is not moving yet) = mgh

 

Gravity, for reasons we do not have to belabor, is what is called a conservative force.  This means that an object under the influence only of gravity, its mechanical energy remains constant.  If we ignore air resistance as the ball falls, then, its total mechanical energy must be conserved, because gravity is the only force working on it.

 

Just before it hits the ground, what is its mechanical energy?

 

ME = GPE + KE = 0 (because the ball is now at zero height) + (1/2)mv² = (1/2)mv²

 

So now the mechanical energy is all kinetic.  But since the total mechanical energy has stayed constant, this kinetic energy must exactly equal the amount of potential energy the ball had at its release above.  As the ball fell, that potential energy was converted to kinetic energy, but the total mechanical energy stayed the same.  It's as if we had a jar of water labeled "potential," and an identical jar labeled "kinetic," which started empty.  The ball falling was the equivalent of pouring the water from our potential to our kinetic jar.  All the water ended up in the kinetic jar, but the total amount of water (like the total amount of mechanical energy) stayed the same, if we did not spill as we poured.

 

Suppose the ball bounced off the ground and started back upward at the same speed it had just before hitting the ground.  Then, its kinetic energy at the ground heading up would be the same as it had been just before it hit, and we would have the falling situation in reverse.  It would rise (as we pour from the kinetic back to the potential jar) with the same total amount of mechanical energy as it had before.  Therefore, it must rise back to the same height and come to a stop, since its potential energy at that position once again equals its total mechanical energy for this entire process.

 

But there is another possibility.  As the ball bounces, the ground exerts a contact force on it (normal force), and the ground is NOT necessarily conservative.  Some of its mechanical energy could be used generating heat through friction, or generating sound waves, or for other processes.  This is like we poured all the water into the kinetic jar, then poured some out of the kinetic jar onto the ground.  That represents the mechanical energy lost to these non-conservative actions.

 

That means there is less water to pour back into the potential jar, so the ball's total potential energy once it comes to a stop after rising from its bounce must be less than the potential energy it started with.  Since the ball's weight is (ostensibly) the same, the lower potential energy must mean the ball does not bounce back as high.  The difference is potential energy between when it was released and when it reaches its maximum height after its bounce equals the mechanical energy lost to heat and sound during the bounce.

 

I think the efficiency they are talking about is the fraction/decimal, or the percent, of mechanical energy retained after the bounce.  We can measure the mechanical energy of both the initial drop and the bounce back by measuring the potential energy when the ball is at its highest point on each leg (the drop on the first leg, and the highest point it comes to on the bounce back), since all the mechanical energy is potential at that point. If the weight of the ball is known, then calculating the potential energy at those points just involves measuring the height.

 

So I would say, just measure the height it is dropped from and the height it returns to.  Then calculate the potential energy at those points.  Those will represent the mechanical energies of the drop and bounce back.  The ratio of the bounce back mechanical energy to the drop mechanical energy represents, I think, the efficiency they ask for.  If it bounces back, for example, to 95% of its initial height, then 95% of its mechanical energy has been retained, and 5% lost to non-conservative actions during the bounce.

 

This is a lot of description, but feel free to ask more questions about it, either here or in an online session.