Question

1. two balls are pushed from a platform down two different ramps as shown: one ramp is much steeper than the other. which ball has the higher velocity at the bottom?

Explanation:

Step1: Consider conservation of energy

Assume no non - conservative forces (like friction) act. The initial potential energy of each ball at the platform is $U = mgh$, where $m$ is the mass of the ball, $g$ is the acceleration due to gravity and $h$ is the height of the platform above the bottom of the ramp.

Step2: Analyze final kinetic energy

At the bottom of the ramp, all of the initial potential energy is converted into kinetic energy $K=\frac{1}{2}mv^{2}$. Since the initial height $h$ is the same for both balls (the height of the platform is fixed), and $U = K$ (by conservation of mechanical energy), $\frac{1}{2}mv^{2}=mgh$. Solving for $v$, we get $v=\sqrt{2gh}$. The mass $m$ cancels out and the velocity $v$ depends only on $g$ and $h$.

Answer:

Both balls have the same velocity at the bottom.