What Are the Final Speeds of a Material Point and a Wedge Without Friction?

[Karozo]

Homework Statement

I think that the image is clear.
There isn't friction, not the material point with wedge, not the wedge with the floor.

At time t=0 the material point start to move, I need to find the final speed of the two objects at time t\rightarrow\infty.

Homework Equations

I have used conservation of energy and momentum.

The Attempt at a Solution

So I have two equation:
\frac{1}{2}m{v_m}^2+\frac{1}{2}M{v_M}^2 = mgR

m{v_m}+M{v_M}=0

And the solution is {v_m}=\sqrt{\frac{mgR}{\frac{1}{2}m+\frac{1}{2}\frac{m^2}{M}}}

Am I wrong?

 

[tms]

It can be simplified, but it looks okay. What about the velocity of the other object?

 

[Karozo]

Well, if V_m is right is very simple to find V_M, so I haven't written it.

 

[Karozo]

I have also a similar problem, you can see the image.

In this the mass m, start with an initial speed v_0, you have to find v_0 so that the material point has maximum height R.

I think that is right to use the two equations:

\frac{1}{2}m{v_m}^2+\frac{1}{2}M{v_M}^2+mgR=\frac{1}{2}m{v_0}^2 energy

m{v_m}+M{v_M}=mv_0 momentum

for the point of maximum height, and then you have {v_m}={v_M} , because the two bodies are in contact.

 

[tms]

I have also a similar problem, you can see the image.

Next time, you should put a new problem in a new thread.

In this the mass m, start with an initial speed v_0, you have to find v_0 so that the material point has maximum height R.

I think that is right to use the two equations:

\frac{1}{2}m{v_m}^2+\frac{1}{2}M{v_M}^2+mgR=\frac{1}{2}m{v_0}^2 energy

m{v_m}+M{v_M}=mv_0 momentum

for the point of maximum height, and then you have {v_m}={v_M} , because the two bodies are in contact.

That appears correct.