How can i calculate the compression or deformation of an object after a collision?

Question

To be more precise about what I want to ask, will an object of 1 kg going at 10 m/s hitting another object of 1000 kg (at rest) have the same deformation as if we switch the velocities and the one of 1000 kg hits the 1 kg object? Whether it's elastic or inelastic collision, will the deformation of the 1 kg object be the same in both cases? Can someone show the equations I need to use?

Answer 1

In the absence of anything else in the universe, these two scenarios are identical. This is because Newton's laws equally hold in any inertial reference frame and one can convert between the two scenarios by changing reference frames.

The question brought up in a comment is not quite the same as OP's question: if we compare a boulder being dropped on your from 5 metres to you falling on a boulder from 5 metres, it is true that the relative velocities at the moment before the collision will be the same because the law of acceleration due to gravity is universal. However, the overall situation is different because (I presume) there is an asymmetry between the boulder resting on the ground and you resting on the ground. When the boulder rests on the ground, you can bounce off of it, with some of the energy going into your kinetic energy. When you rest on the ground, you cannot bounce off of the boulder, so more of the energy goes into deforming you. Yes, the boulder can gain some kinetic energy and bounce off of you, so you can use conservation of energy and momentum to try to figure this out in more detail.

Answer 2

Given Newton's 3rd law, the forces acting on the two bodies are equal and opposite and thus the deflection is strictly a matter of the geometry and elastic properties of the two objects.

The initial conditions are of little consequence here, as the two factors that go into the total impulse $J$ exchanged is the reduced mass $m^\star = \frac{m_1 m_2}{m_1 + m_2}$ and the relative velocity $v^\star = v_1-v_2$

$$ J = 2\, m^\star v^\star $$

which body has which velocity does not matter.

What does matter is how each body responds to the impulse $J$ in terms of changes in momentum, and in terms of elastic response. But the driver here is the common impulse $J$.