Conservation of linear momentum, undergrad particle dynamics

[greg_rack]

Homework Statement

block a has a mass of 2kg and slides into an open ended box B with a velocity of 2m/s. If the box B has a mass of 3kg and rests on a plate P of mass 3kg, calculate the distance the plate slides on the floor before it stops. Also, how long is it after impact before all motion ceases? Coefficients of kinetic friction are given, and so is the statics one between the plate and the floor

Relevant Equations

conservation of momentum, impulse

Hi all,

I'm opening this thread because of my uncertainty in how to correctly approach this exercise.
My first thought was that, since the plate is subject to friction with the floor, it is going to stop, thus the final moment is 0. Hence, from the conservation of linear moment:
$$m_Av_A+\sum \int_{t_1}^{t_2}Fdt=0$$
Now, couldn't we already solve for $t_2$ given the only external force to be the kinetic friction between the plate and the floor?

To compute the distance traveled by the plate, I'm totally disoriented... I believe conservation of energy should be used, by I can't really find a decent way to formalize my ideas.
I realize my thoughts are a complete mess, but it's one of the first exercises I do of the kind, and this one doesn't seem very immediate.

Definitely needing some hints to get on track :)

 

[PeroK]

I assume there is a totally inelastic collision between A and B to get you started.

 

[greg_rack]

I assume there is a totally inelastic collision between A and B to get you started.

Thanks for the hint!
At the end, indeed, it wasn't very immediate... I started by calculated the velocity of block A+B due to the inelastic collision, and then the accelerations of AB and P along the x direction caused to friction, w.r.t to the floor.
After that, I computed the relative acceleration of AB relative to P and thus the time needed for AB to slide on P, which enabled me to calculate distance traveled by P, as long as the velocity of block AB+P.
Then, since sliding has finished, the three blocks will move as a whole, in an easily analyzable accelerated horizontal motion!
Took me quite some time to realize all of this though :)

 

[haruspex]

The problem is not well posed.
First, we have to assume there is no friction between the boxes. Instead, A hits the far end of B inelastically.
Secondly, we are not told the static friction (call if $\mu_s$) between B and P. For sliding between them to commence, the frictional force must reach $(m_A+m_B)g\mu_s$. If that exceeds $(m_A+m_B+m_P)g\mu'_s$ then B will not slide on P at all.

I also note that it asks for "the distance the plate moves after it stops sliding". That, surely, is zero. No doubt they meant the distance it will have moved. I trust it is not a trick question.

Setting that aside, @greg_rack, did you check that the static friction between P and ground will be overcome immediately?

And what answers did you get? I have that P slides $0.3m^2/s^2/g$ before AB stops sliding along P and $0.2m^2/s^2/g$ thereafter.