Momentum transfer of brittle objects

[sbartyc]

Homework Statement

Consider 2 spheres weighing 1 kg each and traveling at 50 m/s toward a wall. The spheres have the same density and presented area but when they hit the wall, sphere A breaks apart and falls to the ground while sphere B falls to the ground but does not break apart. Do the spheres transfer the same amount of energy into the wall? If not, which sphere transfers more energy into the wall?

Relevant Equations

KE=0.5 x mass x velocity^2

I assume KE is conserved therefore, the KE transferred will be the same. But intuitively, this doesn't seem correct. Seems like the object that breaks apart would transfer less energy than the object that doesn't. Any thoughts?

 

[kuruman]

Homework Statement:: Consider 2 spheres weighing 1 kg each and traveling at 50 m/s toward a wall. The spheres have the same density and presented area but when they hit the wall, sphere A breaks apart and falls to the ground while sphere B falls to the ground but does not break apart. Do the spheres transfer the same amount of energy into the wall? If not, which sphere transfers more energy into the wall?
Homework Equations:: KE=0.5 x mass x velocity^2
I assume KE is conserved therefore, the KE transferred will be the same. But intuitively, this doesn't seem correct. Seems like the object that breaks apart would transfer less energy than the object that doesn't. Any thoughts?

Do the spheres bounce back after they hit the wall or do they fall straight down? If they fall straight down, kinetic energy is not conserved.

Consider this thought experiment. Say you drop the sphere from a height of 1 mm on a hard table and it doesn't break, but if you drop it from a height of 1 mile, it does break. Clearly in between 1 mm and 1 mile there is a threshold height $h$ that separates breaking from not breaking. Now consider dropping the ball from height $h-1~ \mathrm{mm}$ (it doesn't break) and height $h+1~ \mathrm{mm}$ (it breaks). In which of the two cases do you think the ball transfers more energy to the table and why?

 

[jbriggs444]

I assume KE is conserved therefore, the KE transferred will be the same.

KE is not conserved.

The idea that any energy at all is transferred to the wall assumes that
1. The wall is not completely rigid. It can deflect in response to an impact. Or
2. The wall is not completely immobile. It can move as a whole in response to an impact. Or
3. Both.

 

[vela]

I assume KE is conserved therefore, the KE transferred will be the same. But intuitively, this doesn't seem correct. Seems like the object that breaks apart would transfer less energy than the object that doesn't. Any thoughts?

You've made a few claims here without providing any sort of justification. Why do you think KE is conserved? Why do you think the KE transferred will be the same in the two cases? Or why not?

 

[haruspex]

The idea that any energy at all is transferred to the wall assumes that
1. The wall is not completely rigid. It can deflect in response to an impact. Or
2. The wall is not completely immobile. It can move as a whole in response to an impact. Or
3. Both.

By conservation of momentum that trio of possibilities is true anyway (and I would include the Earth with the wall system).

@sbartyc , think about the implications of that for energy transfer as the duration of the impact varies.

By the way, I notice the title says momentum transfer, not energy transfer.

 

[sbartyc]

By conservation of momentum that trio of possibilities is true anyway (and I would include the Earth with the wall system).

@sbartyc , think about the implications of that for energy transfer as the duration of the impact varies.

By the way, I notice the title says momentum transfer, not energy transfer.

You're right, I was thinking about momentum transfer when I started the question.
Thinking about the duration of the impact...I don't know if it's different or the same for the 2 spheres. Assuming the wall is not rigid and it can move, I can envision a scenario where impact duration is the same but sphere B causes more displacement of the wall than sphere A (because energy is lost as sphere A breaks apart). This would have to mean that velocity of the system after impact would be higher for sphere B and therefore sphere B would impart more KE into the wall. Correct?

 

[haruspex]

You're right, I was thinking about momentum transfer when I started the question.
Thinking about the duration of the impact...I don't know if it's different or the same for the 2 spheres. Assuming the wall is not rigid and it can move, I can envision a scenario where impact duration is the same but sphere B causes more displacement of the wall than sphere A (because energy is lost as sphere A breaks apart). This would have to mean that velocity of the system after impact would be higher for sphere B and therefore sphere B would impart more KE into the wall. Correct?

I cannot see a clear argument either way.

It might help to consider specific cases. E.g. suppose the force is constant during the impact and treat the wall+earth as some huge mass M. If the duration of the impact is t, you can use conservation of momentum to find the energy transferred to M.
Would the break up case correspond to a lower force over a longer time or a greater force over a shorter time?