# Can we get information of time using energy methods?

I faced up the following problem today:

Two uniform cylinders of the same size but different masses roll down an incline, without slipping, starting from rest. Cylinder A has a greater mass. Which reaches the bottom first?

And he provided solution is:

The initial mechanical energy is all potential energy and hence proportional to mass. When the cylinders reach the bottom of the incline, the mechanical energy consists of both translational and rotational kinetic energy and both are proportional to mass. Since the mechanical energy is constant, the final velocity is independent of mass, so both arrive at the bottom at the same time

I don't entirely buy the argument because the energy method doesn't really give information about time. The mechanical energy is different and, yes, the final velocity independent of mass in each case. But is there really some mechanism that tells us that mass A is not going slower but ends up with the same velocity at the end of the ramp at a time different from mass B that picked velocity faster?

EDIT: Now I'm thinking that it can be argued that the net acceleration on both bodies along the incline is independent of the mass of the bodies. Although now I'm resorting to the kinematics of the inclined plane, and not just relying on energy arguments.

## 1 Answer

If the final velocity is independent of mass, then the final velocity must be the same for both cylinders (since they are otherwise identical).

Because acceleration is constant (forces are constant), then velocity increase is linear.

If velocity increase is linear and both start with equal velocity and end with equal velocity, they must have equal acceleration and hence always have the same velocity. If they always have same velocity, then arrival time is equal.