Question

A girl throws a ball vertically upwards. It takes a time of $3.20s$ to return to her hand.
Assume air resistance is negligible.
What is the initial speed with which the ball is thrown?
(A) $3.07m{s^{ - 1}}$
(B) $7.85m{s^{ - 1}}$
(C) $15.7m{s^{ - 1}}$
(D) $31.4m{s^{ - 1}}$

Answer 1

The velocity of the ball when it reaches its maximum height is zero. The motion is vertically upwards, against the force of gravity.
Formula Used: The formulae used in the solution are given here.
$v = u + at$ where $v$ is the final velocity, $u$ is the initial velocity, $a$ is the acceleration and $t$ is the time taken.

Complete Step by Step Solution: From the question, we have been informed that it takes a time of $3.20s$ for the ball to return to her hand.
Let the distance travelled be $2S$. Thus for the same velocity, the time taken will be half for half the distance travelled. Then, the time that will be taken by the ball to reach maximum height will be $\dfrac{{3.20s}}{2} = 1.60s$.
Let the initial speed be represented by $u$.
By equations of motion, we see that, $v = u + at$ where $v$ is the final velocity, $a$ is the acceleration and $t$ is the time taken.
For vertical motion, the equation is modified to a different form, which states that, $v = u - gt$, where the value of acceleration is equal to , $g$ which is the acceleration due to gravity. It is negative since the body moves vertically upwards, against the force of gravity, that is, in an opposite direction.
We have the values, $t = 1.60s$ and $v = 0$ since the velocity of the ball at its maximum height is always equal to zero. The ball is momentarily at rest at the highest point before it starts its descent towards the ground.
Let the acceleration be $g = 9.8m{s^{ - 1}}$.
Thus, the initial velocity of the ball is,
$0 = u - 9.8 \times 1.60$
$\Rightarrow u = 15.68m{s^{ - 1}}$
The initial velocity is $u = 15.68m{s^{ - 1}} \simeq 15.7m{s^{ - 1}}$.

Hence the correct answer is Option C.

Note: Some assumptions have to be made in answering the question.
First it is assumed that we can ignore air resistance. Thus, this result is only applicable when air resistance is zero.
The vertical motion of a body is controlled by the force of gravity. This means that there is an unbalanced force acting on the ball and so the ball will accelerate downwards. This acceleration is $9.8m{s^{ - 2}} \simeq 10m{s^{ - 2}}$. (the gravitational field strength on Earth).