Two balls are dropped from heights h and 2h respectively from the earth surface. The ratio of time of these ba

Question

Two balls are dropped from heights h and 2h respectively from the earth surface. The ratio of time of these balls to reach the earth is:

Accepted Answer

1.  **Principle:** For a body in free fall starting from rest ($u=0$), the distance fallen ($h$) in time ($t$) is given by the kinematic equation:
    $$h = ut + \frac{1}{2}gt^2 \Rightarrow h = \frac{1}{2}gt^2$$
2.  **Relation:** This implies that time is proportional to the square root of the height:
    $$t = \sqrt{\frac{2h}{g}} \Rightarrow t \propto \sqrt{h}$$ .
3.  **Calculation:**
    *   For the first ball ($h_1 = h$): $t_1 \propto \sqrt{h}$
    *   For the second ball ($h_2 = 2h$): $t_2 \propto \sqrt{2h}$
4.  **Ratio:**
    $$\frac{t_1}{t_2} = \frac{\sqrt{h}}{\sqrt{2h}} = \frac{1}{\sqrt{2}}$$ 
    The ratio is $1 : \sqrt{2}$.

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