Kinetic Energy on Incline Objects: Cylinder, Sphere, Hoop, All Same?

[UrbanXrisis]

Which on the following objects has the least kinetic energy at the bottom of the incline if they have the same mass and radius:
A) cylinder
B) sphere
C) hoop
D) all have the same

I piced B but the answer is D. My question is, the sphere has the least rotational kinetic energy, which means it will roll slowly, making it's kinetic energy less than the hoop or cylinder. I don't understand how they can all have the same kinetic energy when the hoop has a greater rotational kinetic energy.

When a ball is fired from a cannon wth a mass much greater than that of the ball, which of the following is true?

The answer is "The center of mass of the system remains unchanged." I picked "The ball and the cannon have equal momentum after the ball is fired"
I picked this because if the cannon and ball represents an elastic collision. I don't see why my answer isn't valid.

 

[kevinalm]

The first answer is D because the center of mass of each shape dropped the same heighth, so each acquired the same amount of kinetic energy.

Technically, the cannon and the ball don't have the same momentum. Momentum is a vector. But as long as you understand that the center of mass does remain the same you aren't too far wrong. And yes if the cannon and ball conserve momentum, neglecting the gunpowder/gases of course.

 

[UrbanXrisis]

for #1, how can one object gain more velocity than the other while keeping the energy balanced?

for #2, when the cannon fires the ball, the cannon repels backwards keeping the same momentum that it gave the ball. If this was false and momentum was not conserved, then I don't see how the center of mass can be conserved.

 

[kevinalm]

#1 Conservation of Energy. Energy acquired from g*m*delta h = rotational + translational kinetic energy.

#2 The "momentum of cannon and ball are equal" is a trick answer. The magnitudes are equal, yes. But momentum is a vector, and two vectors aren't equal unless both magnitude and direction are equal. These are opposite. So the aren't equal.

The CoM is conserved.