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Two cars of mass $m_{1}$ and $m_{2}$ collide with each other in a completely inelastic collision. So after the collision they continue to go in same velocity. Now suppose I am in one car. So I will consider myself stationary, so my velocity will be $0$. And the other car's velocity relative to me will be $v$.

Now after the collision, I will still consider myself stationary and so my velocity will still be $0$ and the other car's velocity relative to me also will be $0$.

So the total initial momentum $p_{i}$ will be:

$m_{2}v$

And the final total momentum $p_{f}$ will be $0$

So clearly momentum isn't conserved. Clearly I am wrong. Why isn't momentum conserved from my reference frame?

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marked as duplicate by sammy gerbil, John Rennie newtonian-mechanics Aug 9 '18 at 4:43

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You accelerated during the collision, so your reference frame is not inertial. The usual conservation laws only apply in inertial frames, unless you account for the fictitious forces and fictitious work created by the acceleration of the reference frame. Fictitious forces, if not accounted for, can make it look like energy comes from nowhere or disappears, when in reality the energy is being used to accelerate the reference frame.

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  • $\begingroup$ "space" in the system is no longer homogenous, as some positions get hit by a car, and others don't. $\endgroup$ – JEB Aug 8 '18 at 16:08
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But you will feel an impulse,

dp = Fdt

A force is being applied, which results in acceleration, as a result your reference frame will not be an inertial frame of reference (if you somehow manage to stay in a fixed position relative to the car), so momentum will not be conserved in your frame of reference

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  • $\begingroup$ Why won't momentum be conserved in a non inertiall frame?? $\endgroup$ – Theoretical Aug 10 '18 at 7:38
  • $\begingroup$ @Asif Iqubal Because inertial reference frame is a frame which is not accelerating => Non-inertial frame of reference is an accelerating frame. According to the second Law of Newton, F = ma (assume vector notation). p = mv, in classical and non-relativistic case, dp/dt = d(mv)/dt = m(dv/dt) = ma, which means dp = madt, which is not zero, when you integrate over time you will get a non-zero result, so there is a change in total momentum. $\endgroup$ – Adhamzhon Shukurov Aug 10 '18 at 11:38
  • $\begingroup$ If the frame was inertial, the acceleration would be zero so change in total momentum would also be zero, because dp = m*0*dt, if you integrate it over time the result will be zero, so the total momentum is conserved in that frame of reference. $\endgroup$ – Adhamzhon Shukurov Aug 10 '18 at 11:38
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Your rest reference frame has changed by the collision. This is what makes collisions so dangerous. Momentum is different as determined from different reference frames, as it transforms as a vector. In each reference frame, before and after the collision, momentum is conserved.

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