Find Velocity: Easy Method to Solve Complex Physics Problem

[utkarshakash]

Homework Statement

I have attached a picture. The question is in the diagram itself.

Homework Equations

The Attempt at a Solution

I tried to solve it by drawing free body diagrams and writing the equations but that seemed enormous and complex too and it seems that it will take a lot of time to solve it by that method. I know that this problem requires me to apply conservation of momentum and energy conservation principle but how? No doubt it can be solved by the traditional method but it would be too long and calculations will be extremely complex. There must be some easier way to do this.

 

[cmmcnamara]

Can you write the equations for energy at the two states of the system, before and after the slide?

 

[utkarshakash]

If I assume the final velocity of wedge to be v$_{w}$ and of block to be v$_{b}$ wrt ground then using conservation of energy

mgh=(Mv$_{w}$$^{2}$)/2+(mv$_{b}$$^{2}$)/2

Now what to do next?

 

[Saitama]

Now what to do next?

...this problem requires me to apply conservation of momentum...

 

[utkarshakash]

I know that I have to apply conservation of momentum. But I don't even know the velocities. I have only assumed them according to my convenience. Nevertheless if you want me to try using that here's it

Along X-axis(taking left direction as -ve)
$\vec{P}_{initial}$=0
$\vec{P}_{final}$=-Mv$_{w}$+??

Here the problem arises. I don't know what is the velocity of block in the x-direction. How do i proceed?

 

[azizlwl]

The work done by gravity equal to MgSinθ°d
This energy is tranlated to final kinetic energy at the bottom.
Since both objects experience same force(Newton's 3rd Law), the momentum of for each object is equal and opposite.

 

[Saitama]

I know that I have to apply conservation of momentum. But I don't even know the velocities. I have only assumed them according to my convenience. Nevertheless if you want me to try using that here's it

Along X-axis(taking left direction as -ve)
$\vec{P}_{initial}$=0
$\vec{P}_{final}$=-Mv$_{w}$+??

Here the problem arises. I don't know what is the velocity of block in the x-direction. How do i proceed?

Assume that the velocity of block when it reaches the bottom is v_b. Equate the initial and final momentum. You will get a relation between v_b and v_w. Use that relation in the energy equation.