Question

Question: A ball slides up a frictionless ramp. It is then rolled without slipping and with the same initial velocity up another frictionless ramp (with the same slope angle). In which case does it reach a greater height, and why?

Short answer

The height reached by the ball during rolling is greater than during sliding.

Step-by-step solution

Height reached by the ball during sliding.

During sliding of the ball, the translational kinetic energy of the ball is converted to the potential energy.

\(\begin{array}{}K.E{._{trans.}} &= P.E.\\\frac{1}{2}m{v^2} &= mgh\\h &= \frac{{{v^2}}}{{2g}}\end{array}\)

Here, h is the height reached, v is the velocity of the ball, m is the mass of the ball, and g is the acceleration due to gravity.

Height reached by the ball during rolling without friction.

During rolling of the ball without friction, the rotational and translational kinetic energy of the ball is converted to potential energy.

\(\begin{array}{c}K.E{._{trans.}} + K.E{._{rot.}} &= P.E.\\\frac{1}{2}m{v^2} + \frac{1}{2}I{\omega ^2} &= mgh\\\frac{1}{2}m{v^2} + \frac{1}{2} \times \left( {m{r^2}} \right) \times {\frac{v}{{{r^2}}}^2} &= mgh\\m{v^2} &= mgh\\h &= \frac{{{v^2}}}{g}\end{array}\)

Clearly, the height reached by the ball during rolling is greater than during sliding.