Question

If two balls of masses ${m_1}$ and ${m_2}$ are dropped from the same height, then the ratio of the time taken by them to reach ground will
A. ${m_1}:{m_2}$
B. $2{m_1}:{m_2}$
C. $1:1$
D. $1:2$

Answer 1

Hint: Galileo performed a well-known experiment in the church while seated at a particular height. With two balls of various masses in his hand, he simultaneously dropped them from the same height and was amused to see that the two balls, regardless of their individual masses, all fell to the earth at the same time.

Formula used:
Newton’s equation of motion,
$s = ut + \dfrac{1}{2}g{t^2}$
Here, $s$ is the displacement of the particle, $u$ is the initial velocity, $g$ is the acceleration of gravity and $t$ is the time period.

Complete step by step solution:
Here, it is important to note that the mass ratio is not extremely large. Air resistance and other elements that can impact the balls' velocities must be disregarded in this situation. When examining the dropping of a ball from a specific height, we must take into account the fact that the two balls are only affected by gravitational force; in other words, we must only take this force into account.

Now since the stone is dropped the initial velocity will be zero ($\therefore u = 0$). For ball of mass ${m_1}$ equation of motion can be written as,
$ \Rightarrow s = \dfrac{1}{2}gt_1^2$
Since we notice that the time is independent of the mass of the ball. Therefore, when the balls are dropped from a specific height, they all fall to the ground at the same time due to the force of gravity. If two balls are travelling at the same speed, they can only collide at the same time. Hence the ratio of the time taken by them to reach ground will be $1:1$.

Therefore, option C is correct.

Note: Here, we must know why the time taken is the same even when the masses are dissimilar. Equation of motion doesn’t depend on the mass of the body. Also, we neglected the air resistance force because the ratio of masses was small. If they were high, we could have used some difference equation to find out the time.