Succesive collisions between a ball and a wall

[LCSphysicist]

Homework Statement

COnversation of momentum.

Relevant Equations

All below

Ignore the letter a. See b.
I disagree with the variation of the velocity. To me, that need to be 2(Vo+V)
" My point is that initial velocity before striking the wall was Vo towards the left and after the collision, the velocity is Vo + 2V towards the right, thus making the change of velocity 2(Vo + V) and not just 2V. " WIfo.
The question arise because of this solution:

I could give an answer equal to the book seeing that V + v ~ v

Is my point above, about the variation of velocity, right? Or am i missing something?

 

[haruspex]

The change in velocity is 2(V+v), but the solution is finding the change in speed.

 

[andrewkirk]

The question is poorly posed, because (1) it does not specify the period over which the average is to be taken and (2) it uses but does not define $l$. If we guess that $l$ is the distance between the walls then the answer to (a) is correct but then how does $x$ differ from $l$? Perhaps $l$ is supposed to be the initial distance and $x$ the distance at some later time $t$, ie $x = x(t)$.

Perhaps it is asking about the average force on each wall over one complete circuit at a later time $t$ when the average distance between the walls is $x(t)$. In that case the time taken for the circuit will be $2x(t)/v(t)$ where $v(t)$ is calculated by adding to $v(0)$ an amount that is $2V$ times the number of circuits completed since time 0 and $x(t)$ is $l-Vt$. To work that out we need to work out how many circuits will have been completed by time $t$, which will depend on the relation between $l$ and $x$.

Also, as haruspex pointed out, your analysis of the change in velocity is out by $2v$, as your calculation is of a change in speed, not of velocity.