Momentum conservation: block-wedge problem

[Krushnaraj Pandya]

Homework Statement

A block of mass m slides down a wedge of mass M and inclination theta from rest. All the surfaces are smooth. Find the speed of the wedge when the speed of the block w.r.t to wedge is v.

Homework Equations

V(c.m.)=m1v1+m2v2/(m1+m2)

The Attempt at a Solution

Conserving momentum in horizontal direction - I got the speed of the wedge as -mscos(theta)/M where s is the velocity of block w.r.t ground. I can't figure out how to incorporate relative velocity in this formula and get it in terms of v

 

[Nathanael]

I got the speed of the wedge as -mscos(theta)/M where s is the velocity of block w.r.t ground.

This isn’t correct. Since the wedge is also moving, s is not directed along θ.

You could break s up into s_x and s_y then say V_wedge =s_xm/M ... but s_x is not scosθ

I’m thinking we should be able to avoid such subtleties by ignoring s and working directly with v.

If we take the instantaneous-frame of the wedge, the block moves at speed v and along the angle θ. Analyzing momentum in this frame seems fruitful.

 

[Krushnaraj Pandya]

This isn’t correct. Since the wedge is also moving, s is not directed along θ.

You could break s up into s_x and s_y then say V_wedge =s_xm/M ... but s_x is not scosθ

I’m thinking we should be able to avoid such subtleties by ignoring s and working directly with v.

If we take the instantaneous-frame of the wedge, the block moves at speed v and along the angle θ. Analyzing momentum in this frame seems fruitful.

I realize my mistake in assuming s in x direction as scos(theta) and the mistakes with my approach...I got the speed of the wedge as mvcos(theta)/M+m, beautiful solution- thank you

 

[Nathanael]

I realize my mistake in assuming s in x direction as scos(theta) and the mistakes with my approach...I got the speed of the wedge as mvcos(theta)/M+m, beautiful solution- thank you

Well done! You are quick.

Just a tip, you should type it as mvcos(theta)/(M+m) to avoid confusion.

 

[Krushnaraj Pandya]

Well done! You are quick.

Just a tip, you should type it as mvcos(theta)/(M+m) to avoid confusion.

Thanks :) I'll keep that in mind from now on