Why is the Velocity of Mass m Negative in the Solution?

[flyingpig]

Homework Statement

[PLAIN]http://img522.imageshack.us/img522/8308/deskqd.jpg

Solutions

[PLAIN]http://img542.imageshack.us/img542/6941/32144235.jpg

The Attempt at a Solution

I have two problems

For part b, shouldn't the mass m have a positive velocity v₀ since it is going counter clockwise? How do we know that v₀ for mass m is negative? Because my answer was the same except by a negative sign...

For part c, when they say the total mech energy, after collision, why is that different from the kinetic energy after the collision?

Here is what I did

$$\stackrel{1}{2}mv_{0}^2 + \stackrel{1}{2}(3m)v_{0}^2 = \stackrel{1}{2}(3m + m)v'^2$$

$$2mv_{0}^2 = 2mv'^2$$

$$v_{0}^2 = v'^2$$

$${\sqrt{\stackrel{GMe}{2Re}} }^2={ \stackrel{1}{2}\sqrt{\stackrel{GMe}{2Re}}^2}$$

$$\stackrel{GMe}{2Re} = \stackrel{1}{4}\stackrel{GMe}{2Re}$$

$$\stackrel{GMe}{2Re} = \stackrel{GMe}{8Re}$$
I calculated the loss of kinetic energy after the collision or the heat gained by friction? If I were to approach it like that, how should I go to get the answer in the Key?

Thanks

 

[anathema]

.
Thank you for your post and for your efforts in solving the problem. I will address your two questions separately.

For part b, it is important to consider the direction of the velocity in relation to the chosen coordinate system. In this case, the velocity v₀ of mass m is negative because it is moving in the opposite direction of the positive direction chosen for the coordinate system. This is why it is important to be consistent with your coordinate system throughout the problem. If you had chosen the opposite direction as positive, then v₀ would be positive as well.

For part c, the total mechanical energy after the collision is different from the kinetic energy after the collision because mechanical energy includes both kinetic and potential energy. In this case, the potential energy is due to the gravitational interaction between the masses and is not conserved during the collision. Therefore, the total mechanical energy will change after the collision, but the kinetic energy alone will not be conserved.

Your approach to solving the problem is correct, but your calculations may have some errors. I would suggest going through your steps again and double checking your algebra to see if you made any mistakes. It is also important to note that the loss of kinetic energy after the collision is due to both friction and the change in potential energy, so you will need to consider both factors in your calculations.

I hope this helps clarify your questions. Keep up the good work in your studies!