Question

When a ball is dropped from a height $h$ it takes $t$ sec to reach the ground. If the same experiment is done on a different planet having the mass 100 times the earth’s mass and radius 10 times the earth’s radius, then the time it will take to cover the same height $h$ is

• A. $t$
• B. $100t$
• C. $\frac{t}{100}$
• D. $\frac{t}{10}$

Hint:

The acceleration due to gravity $g$ depends on the mass and radius of a planet as $g = \frac{GM}{R^2}$. When comparing gravitational effects on different planets, always compute $g$ first to determine its impact on motion.

Answer 1

The Correct Option is A

The time taken for a ball to fall from a height $h$ under gravity is determined using the equation of motion $h = \frac{1}{2}gt^2$, where $g$ is the acceleration due to gravity and $t$ is the time.
- On Earth: Let the acceleration due to gravity be $g_{\text{earth}}$, the mass of Earth be $M$, and the radius of Earth be $R$. The time to fall height $h$ is $t$. Using the equation:
\[ \begin{align} h = \frac{1}{2} g_{\text{earth}} t^2 \implies t^2 = \frac{2h}{g_{\text{earth}}} \implies t = \sqrt{\frac{2h}{g_{\text{earth}}}} \] - On the other planet: The planet has mass $M' = 100M$ and radius $R' = 10R$. The acceleration due to gravity $g$ is given by $g = \frac{GM}{R^2}$, where $G$ is the gravitational constant. For the planet: \[ \begin{align} g_{\text{planet}} = \frac{G M'}{(R')^2} = \frac{G (100M)}{(10R)^2} = \frac{100 G M}{100 R^2} = \frac{G M}{R^2} = g_{\text{earth}} \] The acceleration due to gravity on the planet is the same as on Earth, $g_{\text{planet}} = g_{\text{earth}}$. Now, calculate the time $t'$ to fall the same height $h$ on the planet: \[ \begin{align} h = \frac{1}{2} g_{\text{planet}} (t')^2 \implies (t')^2 = \frac{2h}{g_{\text{planet}}} = \frac{2h}{g_{\text{earth}}} \implies t' = \sqrt{\frac{2h}{g_{\text{earth}}}} = t \] Since $g_{\text{planet}} = g_{\text{earth}}$, the time $t'$ is equal to $t$. Thus, the time to fall the same height $h$ on the planet is $t$, which matches option (1).
Thus, the correct answer is (1).