Question: Two balls of equal masses are thrown upwards, along the same vertical direction at an interval of \(...

Question

Two balls of equal masses are thrown upwards, along the same vertical direction at an interval of $2$ seconds, with the same initial velocity of $40\,m/s$ . Then these collide at a height of ? ( ${\text{take }}g = 10m{\text{ }}{s^{ - 2}}$ )
A. $120\,m$
B. $75\,m$
C. $200\,m$
D. $45\,m$

Answer 1

Solution

This problem is based on simple one-dimensional motion. The two balls are thrown in the same line one after another at an interval of 2 seconds so they collide after some time at a height. One ball is in the way of returning and the second one is in the way of moving upwards. To calculate the height, we need to use the equation of motion.

Formula used:
$S = ut + \dfrac{1}{2}a{t^2}$
Here, $S$ is the total displacement, $u$ is the initial velocity, $a$ is the acceleration and $t$ is the total time.

Complete step by step answer:
Suppose $t$ is the time taken to reach the height $H$ and acceleration is the acceleration due to gravity.
For the 1st ball,
$H = 40t - \dfrac{1}{2}g{t^2}$
For the second ball,
$H = 40(t - 2) - \dfrac{1}{2}g{(t - 2)^2}$
So, by comparing both the equation we get,
$40t - \dfrac{1}{2}g{t^2} = 40(t - 2) - \dfrac{1}{2}g{(t - 2)^2}$
$\Rightarrow 2gt = 100 \\\ \Rightarrow t = \dfrac{{100}}{{20}} \\\ \Rightarrow t = 5\operatorname{s} \\\$
Therefore, the height is,
$H = 40 \times 5 - \dfrac{1}{2} \times 10 \times 25 \\\ \therefore H= 75m \\\$
Hence the correct option is B.

Additional information: In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables.

Note: Here, we consider the direction also while solving this problem. The acceleration due to gravity is negative because downward direction is negative while the upper direction is taken positive therefore the height is treated as a positive quantity and the velocity of projection is also considered positive since it has been projected upwards.