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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?

Qmechanic
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    What have you tried? Where did you get stuck? – Quantum Mechanic Dec 29 '21 at 15:41
  • @QuantumMechanic it's just an idea in my head i want to clarify –  Dec 29 '21 at 15:48
  • @QuantumMechanic So what i think is that when the bigger object hits the smaller, the smaller will have more deformation than if it happen otherwise. –  Dec 29 '21 at 15:53
  • You fall $5,\rm m$ onto a large boulder and break a leg compared with the large boulder falling $5,\rm m$ and then hitting you with the result that you . . . . . . . .? – Farcher Dec 29 '21 at 15:59

2 Answers2

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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.

Quantum Mechanic
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  • Sorry but still i can't get this, the kinetic energy in the collision gonna be more when the bigger mass hits the smaller. So how can the smaller have the same deformation in both cases? –  Dec 29 '21 at 22:09
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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 $J$.

John Alexiou
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  • So, will the smaller object have the same deformation or compression bouncing in the bigger one, as if the bigger one hits the smaller? What about the kinetic energy? –  Dec 29 '21 at 23:18
  • I did not say the same. Deformation is a function of geometry and materials. Typically the bigger an object the less stiff it is, and therefore it would have more deformation but that is only if the two objects are made of the same material and have similar shapes. What about KE? – John Alexiou Dec 30 '21 at 01:49
  • Despite the material the smaller mass is made off, the bigger one is made by very strong material. How can the deformation of the smaller mass be the same if the bigger one hits the smaller as if the smaller bounces in the bigger? –  Dec 30 '21 at 11:27
  • The smaller after bouncing in the bigger, gonna get returned it same KE by the law of conservation of energy. The bigger one will give the smaller more KE energy after hiting it. If KE is different in the two cases, and the material of the smaller resists its KE but not 2KE which the bigger gives him, How can the deformation be the same if the KE is twice bigger? –  Dec 30 '21 at 11:39