Question

A ball is thrown vertically upwards with a velocity of $20\,m{s^{ - 1}}$ from the top of a multi storey building. The height of the point where the ball is thrown $25\,m$ from the ground. How long will it be before the ball hits the ground? Take $g = 10\,m{s^{ - 2}}$ .
(A) $t = 5\,s$
(B) $t = 10\,s$
(C) $t = 15\,s$
(D) $t = 20\,s$

Answer 1

Use the equation of the motions formula to find the following. Find the time taken for the upward movement of the stone and then find the maximum height of the stone from the building. Then find the time taken by the stone to reach the ground adding both the time provides the answer for time taken.

Useful formula:
The equation of the motions are given by
(1) $v = u + at$
(2) $s = ut + \dfrac{1}{2}a{t^2}$
Where $u$ is the initial velocity of the ball, $v$ is the final velocity of the ball, $a$ is the acceleration, $t$ is the time taken and $s$ is the distance travelled by the ball.

Complete step by step solution:
It is given that the
The velocity at which the ball thrown up, that is initial velocity, $u = 20\,m{s^{ - 1}}$
Height of the building, $h = 25\,m$
The acceleration due to gravity, $g = 10\,m{s^{ - 2}}$
Using the first equation of the motion,
$v = u + at$
Substituting the known values,
$0 = 20 + \left( { - 10} \right)t$
By performing the basic arithmetic operation, we get
$t = 2\,s$
By using the second equation of motion,
$s = ut + \dfrac{1}{2}a{t^2}$ -----------(1)
In order to find the maximum distance covered,
$s = 20 \times 2 + \dfrac{1}{2} \times - 10 \times {2^2}$
By simplification of the above equation, we get
$s = 20\,m$
The above distance represents the height at which the stone flies from the building. Hence the total height at which the stone flies is calculated by adding both the heights.
$h = 20 + 25 = 45\,m$
In order to calculate the total time taken to hit the ground, substitute the total height in the equation (1)
$s = ut + \dfrac{1}{2}a{t^2}$
$45 = 0 + \dfrac{1}{2}10{t'^2}$
By performing basic arithmetic operations, we get
$t' = 3\,s$
Hence the total time, $t = t + t' = 2 + 3 = 5\,s$

Thus the option (A) is correct.

Note: The acceleration of the stone while moving upward is taken as $- 10$ this is because the work of throwing the stone upward is done against the gravitational force. So $a = - g$ . The acceleration of the tone while moving down is $10$ , because the velocity of stone is zero and its movement is only due to the acceleration due to gravity.