Question

An object is dropped from rest at a height of 150 m and simultaneously another object is dropped from rest at a height of 100 m. What is the difference in their heights after 2 seconds if both the objects drop with the same accelerations? How does the difference in heights vary with time?

Answer 1

Given:

The height of the first object, $h_1=150m$

The height of the second object, $h_2=100m$

Finding the height $d_1$ of the first object after 2 seconds:

Let $s_1$ be the distance covered by the first body in 2 seconds. Then,

$d_1=h_1–s_1$

by using the equation of motion, $s=ut+\frac{1}{2}g{t}^2$

$$\begin{gathered} {\mathrm{s}}_1=0\times 2+\frac{1}{2}{\mathrm{gt}}^2 \\ \mathrm{As}\mathrm{u}=0\mathrm{for}\mathrm{a}\mathrm{freely}\mathrm{falling}\mathrm{object} \\ \Rightarrow {\mathrm{s}}_1=\frac{1}{2}{\mathrm{gt}}^2 \\ \mathrm{Similarly},{\mathrm{s}}_2=\frac{1}{2}{\mathrm{gt}}^2 \\ \mathrm{Now},\mathrm{finding}\mathrm{the}\mathrm{height}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{object}\mathrm{using},{\mathrm{d}}_1={\mathrm{h}}_1–{\mathrm{s}}_1 \\ {\mathrm{d}}_1=150–\left(\frac{1}{2}{\mathrm{gt}}^2\right) \\ {\mathrm{d}}_1=150–(\frac{1}{2}\times 10\times 4) \\ {\mathrm{d}}_1=150–20 \\ {\mathrm{d}}_1=130\mathrm{m} \end{gathered}$$

Therefore the height of the first object after 2 seconds is 130 m.

Finding the height $d_2$ of the second object after 2 seconds:

Let $s_2$ be the distance covered by the second body in 2 seconds. Then,

$d_2=h_2–s_2$

$$\begin{gathered} {\mathrm{d}}_2=100–(\frac{1}{2}{\mathrm{gt}}^2) \\ {\mathrm{d}}_2=100–(\frac{1}{2}\times 10\times 4) \\ {\mathrm{d}}_2=100–20 \\ {\mathrm{d}}_2=80\mathrm{m} \end{gathered}$$

Therefore, the height of the second object after 2 seconds is 80 m.

Finding the difference between the heights of the two objects:

$$\begin{gathered} {\mathrm{d}}_1–{\mathrm{d}}_2=130\mathrm{m}–80\mathrm{m} \\ {\mathrm{d}}_1–{\mathrm{d}}_2=50\mathrm{m} \end{gathered}$$

So, the difference is the same, i.e. 50 m.

This concludes that the difference in height of the two objects does not depend on time and will always be the same.