Bumps on a Track

QUESTION:

Two smooth tracks of equal length have "bumps" of the same curvature. The only difference is that A bumps ^up and B bumps _down.

drawing of balls on two tracks with bumps

If two balls start simultaneously with the same initial speed, the ball to complete the journey first is

Ball A.
Ball B.
both balls take the same time.

EXTRA:

If the initial speed = 2 m/s and the speed of the ball at the bottom of the curve on Track B is 3 m/s, then the speed of the ball at the top of the curve on Track A is

1 m/s.
> 1 m/s.
< 1 m/s.

ANSWERS: B and F

Although both balls have the same speed on the level parts of the tracks, the speeds along the curved parts differ. The speed of the ball everywhere along Curve B is greater than the initial speed. Everywhere along Curve A, it is less. So the ball on Track B finishes first.

Does the gain in speed at the bottom of Curve B equal the loss at the top of Curve A? No! Speed isn't conserved: energy is. The loss in kinetic energy at the top of A will be equal to the gain in kinetic energy at the bottom of B…if there is enough energy to begin with.

There isn't: The initial KE [1/2 m2²] is less than the gain in KE at the bottom of B [1/2 m(3² – 2²)]. At 2 m/s, the ball will not even make it to the top of A's curve.

Back to question