What Determines the Direction of Motion in a Plank and Spring System?

[Rikudo]

Homework Statement

A block of mass m is placed on a plank of mass M that rests on a frictionless floor. The block is connected to an obstruction on the plank with the help of a light spring of stiffness k as shown in the figure . Coefficient of friction between the plank and the block is μ. Initially when the spring is relaxed , the block is given a velocity Vo towards the obstruction . If the block stops on the plank before reversing its direction of motion relative to the plank , find the velocity Vo . Acceleration due to gravity is g.

Relevant Equations

Momentum conservation
Work - Energy theorem (?)

I don't understand the question. If it want the mass to stop and reverse its direction, then does that not means $vo$ can be anything? (obviously not 0 since it will make the system not moving at all).

 

[haruspex]

If it want the mass to stop and reverse its direction, then does that not means $vo$ can be anything?

Could there be some range of velocities for which it does not reverse direction?

 

[Lnewqban]

Consider that initial velocity Vo is measured respect to both, the plank and the floor.
By the time the block stops respect to the plank ("the block stops on the plank before reversing its direction of motion"), the plank has acquired certain velocity respect to the floor; therefore, its velocity respect to the floor must match that certain plank's velocity.

 

[Rikudo]

When they are in motion, the mass's speed will decrease while the plank's will increase. Eventually, they will have the same speed (which means at this instant, the mass is not moving with respect to the plank). No matter what Vo is, they will undergo this event.

 

[haruspex]

No matter what Vo is, they will undergo this event.

That is for the event that the block comes to rest relative to the plank. I asked about the event that the block subsequently moves in the other direction relative to the plank.

 

[Lnewqban]

Consider as well that the sliding block has two simultaneous ways to transfer part of its initial KE to the plank: constant force of friction and variable elastic force of the spring, both reducing its velocity.

 

[Rikudo]

Now I understand. Thank you, all of you!