Why Do Objects Bounce? Exploring Momentum and Energy Conservation

[Dark85]

TL;DR

Proof of why objects bounce using conservation of momentum and energy

Consider a body of mass m held above a rigid surface at a height h. Let the surface have such a large mass that any force exerted by the body during this entire scenario practically does not move the surface. This implies that before the body hits the surface and after hitting it, the momentum of the surface is zero throughout. Let the body, after hitting the surface, attain a final momentum of mv Since the body is initially held at rest, it's initial momentum is zero. Since there is no external force on the body-surface system, momentum is conserved i.e. :
initial momentum = final momentum
mu + 0 = mv + 0
0 = mv
v = 0 m s^-1
This implies the final possible state can be at rest on the surface or at rest after bouncing to any height, including the initial height h(We assume there is no dissipation of energy as heat or sound)

But if it is at rest on the surface, it implies kinetic energy is 0 J. Since we assumed that no energy is dissipated, it implies energy is not conserved hence this final state is not possible. It is the same for all other states where the ball bounces to a smaller height because in this case also energy is not conserved as the kinetic the initial energy mgh is not equal to the final energy mg*(a height smaller than h). Only if the ball bounces back to the same height h is when momentum and energy is conserved. Hence, this why an object bounces rather than just hitting the surface and staying at rest.

 

[kuruman]

What causes the mass above the surface to start moving as soon as it's released from rest?
What does Newton's third law have to say about this?
The momentum of the object alone is not conserved because the external force of gravity acts on it.

 

[Dale]

This is nice work. It is clear that you are thinking mathematically. Using infinite mass objects requires some care:

This implies that before the body hits the surface and after hitting it, the momentum of the surface is zero throughout.

Not really. It implies that the velocity is infinitesimal. But an infinitely massive object can have any finite momentum with an infinitesimal velocity.

initial momentum = final momentum
mu + 0 = mv + 0

Because of the above, this equation is not correct. The rest of the argument falls apart here.

In general, when trying to correctly use the idea of an infinitely massive or massless object, the correct procedure is to assume that it has a finite non-negligible mass, do the calculations with that mass, and then take the limit as the mass goes to the relevant extreme.

In this case, if we let the surface have a large mass $M$, and take the inertial frame to be the one where $M$ is initially at rest, then conservation of momentum gives us $$m\ u_m+0=M \ v_M + m\ v_m$$ and conservation of KE gives us $$\frac{1}{2}m\ u_m^2+0=\frac{1}{2}M \ v_M^2 + \frac{1}{2}m\ v_m^2$$ which can be jointly solved to obtain $$v_m=\frac{m\ u_m-M\ u_m}{m+M}$$$$v_M=\frac{2 m\ u_m}{m+M}$$

Now, at this point we can take the limits as $M$ goes to infinity to get $v_m=-u_m$ and $v_M=0$. Importantly, $v_m\ne 0$. Note also, that in this same limit, the momentum of $M$ goes to $M\ v_M=2 m\ u_m$ and its KE goes to $\frac{1}{2} M \ v_M^2 = 0$

 

[Dark85]

Thank you for pointing out my mistakes sir. I will redo it once more till i get a clear understanding. Once again, thank you.

 

[A.T.]

Only if the ball bounces back to the same height h is when momentum and energy is conserved. Hence, this why an object bounces rather than just hitting the surface and staying at rest.

Momentum is a vector that has magnitude and direction. Bouncing back at the same speed but opposite direction does not conserve the momentum of the ball.

 

[Ian Telcher]

TL;DR Summary: Proof of why objects bounce using conservation of momentum and energy

Consider a body of mass m held above a rigid surface at a height h. Let the surface have such a large mass that any force exerted by the body during this entire scenario practically does not move the surface. This implies that before the body hits the surface and after hitting it, the momentum of the surface is zero throughout. Let the body, after hitting the surface, attain a final momentum of mv Since the body is initially held at rest, it's initial momentum is zero. Since there is no external force on the body-surface system, momentum is conserved i.e. :
initial momentum = final momentum
mu + 0 = mv + 0
0 = mv
v = 0 m s^-1
This implies the final possible state can be at rest on the surface or at rest after bouncing to any height, including the initial height h(We assume there is no dissipation of energy as heat or sound)

But if it is at rest on the surface, it implies kinetic energy is 0 J. Since we assumed that no energy is dissipated, it implies energy is not conserved hence this final state is not possible. It is the same for all other states where the ball bounces to a smaller height because in this case also energy is not conserved as the kinetic the initial energy mgh is not equal to the final energy mg*(a height smaller than h). Only if the ball bounces back to the same height h is when momentum and energy is conserved. Hence, this why an object bounces rather than just hitting the surface and staying at rest.

Quit simple, it's just compression and release. Conetic energy makes compression, and stored energy is then released.

 

[Herman Trivilino]

TL;DR Summary: Proof of why objects bounce using conservation of momentum and energy

But if it is at rest on the surface, it implies kinetic energy is 0 J. Since we assumed that no energy is dissipated, it implies energy is not conserved hence this final state is not possible.

The objects have to deform on impact. Even if you assume the deformation of the surface is neglbible, the bouncing object must deform on impact, storing easitic potential energy. After the collision that potential energy is converted to kinetic energy.

If you assume no energy is "dissapated", which simpy means no mechanical energy is converted to internal eneg energy, then you have the idalisation called a perfectly elastic collision.