Usage of momentum conservation during collision of 2 blocks 1 connected to a spring (at rest) and other hitting with a velocity

Question

suppose I have 2 blocks say A and B which are connected by spring and at rest, and there is a third block C which is colliding with A, will it be correct to use momentum conservation equation on this event.

I am confused because, when the block A is colliding, mean to say when the process is ongoing, the A is gaining velocity (accelerating), as a result it exerts force on the spring and the spring exerts the same amount of force back on the block. So there is an external force acting on it, which is discouraging me to use conservation equation.

So if this is a valid reason, what strategy should i use to find out the velocity of the block A?

Answer 1

Energy is conserved as long as you consider the spring as part of the system. At a given point after the collision you will have kinetic energy of the three blocks plus the potential energy of the spring. This would give you the information needed (together with momentum conservation) to find out what you need.

Answer 2

If this is an elastic collision, the problem is quite complex. Conservation of momentum always works - even for inelastic collisions - but in the system you describe, the final velocity of A is not 'fixed'; you can see this by first assuming that the impact takes an infinitesimally short time. In that case, A has not "started to move" by the time the collision is finished, so there is no deformation of the spring and the only force that A feels is the force of the collision with C. In that sense you can treat the collision between A and C without concerning yourself with either B or the spring.

But once the collision is over, A starts moving with some velocity - and then it will compress the spring, and the resulting force will transfer momentum from A to B. After compressing, the spring will start to expand again - in the center of mass frame, A and B will oscillate about the center of mass.

Obviously, if the collision takes a finite amount of time, this oscillatory motion will have started; for this reason you need to make (explicitly state) an assumption about the duration of the initial collision in order to answer the question about the velocity of A.