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Suppose there is an elephant, moving at 10 m/s, in a particular direction. From the opposite direction, a mosquito flies towards the elephant, also at 10m/s, and collides with it. What happens if the collision is elastic? ( Basically, the consequent movement of the two bodies, keeping in mind that the huge difference in mass)

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  • $\begingroup$ What do you think will happen? $\endgroup$ Jun 20, 2017 at 8:50
  • $\begingroup$ I can't be sure. It is mainly due to the fact that it is an elephant colliding with a teensy mosquito. Obviously the elephant will not bounce back. Plus, I don't even see how the coefficient of restitution can be 1 in this case. Basically, I don't know what to think. $\endgroup$
    – TheFool
    Jun 20, 2017 at 8:56
  • $\begingroup$ Get the general velocity formulae for two masses colliding and study their behavior as one mass becomes considerably lesser than the other. $\endgroup$ Jun 20, 2017 at 9:01
  • $\begingroup$ youtube.com/watch?v=2UHS883_P60 You could also refer to this as it is a somewhat similar scenario. $\endgroup$ Jun 20, 2017 at 10:22

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You could consider this collision in the zero momentum frame of the collision. As the mass of the elephant is much greater than the mass of the mosquito,in the zero momentum frame the elephant can be considered to be stationary, and the mosquito moves at $20ms^{-1}$ relative to the elephant. As this is an elastic collision and we consider the mass of the elephant to be infinite, the mosquito bounces off the elephant with speed $20ms^{-1}$ relative to this frame of reference.

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  • $\begingroup$ So the mosquito moves at 30 m/s with respect to an observer? How/Why does it gain kinetic energy, when the collision is elastic? $\endgroup$
    – TheFool
    Jun 20, 2017 at 10:02
  • $\begingroup$ In an actual scenario there would be a very tiny decrease in the momentum of the elephant. However, due to the large difference in the mass, the change in momentum of the elephant is negligible, but it is what supplies the mosquito with the increased kinetic energy. $\endgroup$ Jun 20, 2017 at 10:21
  • $\begingroup$ What do you mean by "zero momentum frame" ? A frame doesn't have any mass, so in my opinion, using the word momentum for frames is inappropriate. Non-inertial frame makes more sense. $\endgroup$
    – Mitchell
    Jun 20, 2017 at 11:01
  • $\begingroup$ By zero momentum frame, I mean an inertial frame of reference where the momentum of the system appears to be zero, that is, a frame moving with respect to the ground at some velocity that is to be determined. $\endgroup$ Jun 20, 2017 at 11:09
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Momentum will be transferred as usual, but since momentum is given by: $$ p = mv $$ and the mosquito's mass is many times smaller than that of the elephant, the resulting loss of velocity by the elephant will be negligible, whilst assuming elastic collision, the mosquito will bounce off the elephant with a considerably larger velocity than before.

The elephant will probably keep walking along at $10 m/s$, since the only requirement of an elastic collision is that energy is conserved. And the elephant will barely lose any kinetic energy by colliding with a mosquito.

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As the mass of the elephant significantly larger than the mosquito, we can assume safely that the velocity of the elephant remains unchanged.

Now,let $v$ be the velocity of the mosquito with respect to the elephant. $v=20$ m/s. Applying conservation of linear momentum and that the co-efficient of restitution equals $1$, we can easily see that the mosquito bounces back with a velocity of $20$ m/s.

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