Why can't we consider relative velocity in two particle system for applying conservation of momentum

Question

In the problem where a man is walking on a platform kept over ice why can't we use the relative velocity of the man with respect to the platform in the conservation of linear momentum.

For example, if a man (mass $m_1$) is walking with velocity $u$ and the recoil velocity of platform (mass $m_2$) is $v$, then on applying the conservation of linear momentum, is it wrong to say

$0 = m_1 u-m_2 v$

(initial momentum=final momentum. Minus sign because the direction of man and platform is opposite of each other)

Answer 1

Conservation laws are strongly dependent on what reference frame you are working in. A reference frame consists of everything one single observer would measure. Any physical situation can always be described by one observer who considers himself stationary. When writing down conservation of linear momentum, you should only consider the velocities that particular observer would see.

In this problem, the velocity $v$ of the plate is given with respect to the ground. This means there is an observer on the ground who measured its velocity. However, the velocity $v$ of the man is given relative to the platform, so there is a second observer on the platform who measures the velocity of the man.

When applying conservation laws, you need to convert all available values to one reference frame (as would be measured by one of the available observers). You cannot use the velocities $u$ and $v$ as given because they were measured in different reference frames. If you want to use the velocity $u$ of the man relative to the plate, you are working in a reference frame where the plate is stationary (because $u$ was measured by an observer moving with the plate who considered himself stationary). In this frame, the ice would be moving instead. So you would have to balance out the momentum from the ice and the man, instead of the momentum from the plate and the man. However, the mass of the ice is not given, so this is not possible.

The sensible solution then, is to convert the velocity $u$ as measured by the moving observer to a velocity $u'$ as it would be measured by the observer on the ice.

TL; DR: Conservation laws are only valid when all involved quantities are measured in the same reference frame (by the same observer). This is not the case in the present problem

I hope this was clear. I know the concept of reference frames and the switching between them was not trivial to grasp when I first encountered it.

Answer 2

First thing: As Bart W pointed out, Conservation Laws are valid in an frame of reference. You can't mix two velocities measured in two different reference frames. Velocities have to be measured in one reference frame like in solution it is measured with respect to ground.

Second Thing: Suppose you are sitting in a frame of reference which moves with velocity of boat at every instant, you must know that the frame you chose now IS A NON-INERTIAL FRAME because boat was at rest with respect to ground and then suddenly gained some velocity that means it started accelerating and so our frame also will start accelerating as it is moving exactly same as boat does so as to keep boat at rest. So now when you consider the system BOAT + MAN you must know that they will experience an external PSEUDO FORCE or INERTIAL FORCE because observations are now carried out in a NON INERTIAL FRAME. So momentum will not be conserved in this new frame. But you can still work out things correctly if you use IMPULSE MOMENTUM THEOREM, which is just generalization of momentum conservation when the system has external forces.

Similarly if you choose a reference frame which moves with velocity of man at every instant, similar arguments will be applied.

But to avoid such little complications why not sit in ground frame and do the calculations easily.