Question

A batsman hits a ball of mass \[0.15\,{\text{kg}}\] straight in the direction of the bowler without changing its initial speed of \[12\,{\text{m}}{{\text{s}}^{ - 1}}\]. If the ball moves linearly, then the impulse imparted to the ball is:
A. \[1.8\,{\text{Ns}}\]
B. \[2.8\,{\text{Ns}}\]
C. \[3.6\,{\text{Ns}}\]
D. \[4.2\,{\text{Ns}}\]

Answer 1

Hint: First of all, we will calculate the initial and the final momentum of the ball with appropriate sign convention. Then we will find out the change in momentum. After that we will substitute the required values and manipulate to obtain the result.

Complete step by step answer:
In the given question, we are supplied the following data:
Mass of the ball is \[0.15\,{\text{kg}}\] .
Speed of the ball is \[12\,{\text{m}}{{\text{s}}^{ - 1}}\] after the batsman hits the ball.
The ball moves linearly.
We are asked to find the impulse imparted to the ball.
To begin with, we will first find the initial momentum of the ball. After the bat hits the ball, the direction of the ball is reversed and due to this the direction of velocity gets reversed. As the physical quantity called momentum is directly dependent on the direction of velocity, the direction of the momentum also gets reversed. So mathematically we can write:
The initial momentum of the ball before hitting the bat, is:
\[{P_1} = m\vec v\] …… (1)
Where,
\[{P_1}\] indicates the initial momentum.
\[m\] indicates the mass of the ball.
\[\vec v\] indicates the velocity vector.
Again, the momentum of the ball after hitting the bat, is:
\[{P_2} = - m\vec v\] …… (2)
Where,
\[{P_2}\] indicates the initial momentum.
\[m\] indicates the mass of the ball.
\[\vec v\] indicates the velocity vector, but in the reversed direction, as after hitting the bat, the direction of the ball gets reversed. So, the negative sign does its job.
Now, we calculate the change in momentum which is the difference between the final momentum and the initial momentum. Impulse is also defined as the change in momentum of an object. Mathematically, it which be written as:
\[\Delta P = \left| {{P_2} - {P_1}} \right|\] …… (3)
Now, we substitute the equations (1) and (2), in the equation (3), and we get:
$\Delta P = \left| {{P_2} - {P_1}} \right| \\$
$\implies \Delta P = \left| { - m\vec v - m\vec v} \right| \\$
$\implies \Delta P = \left| { - 2m\vec v} \right| \\$
$\implies \Delta P = 2mv $ …… (4)
Now, we substitute the required values in the equation (4), and we get:
$\Delta P = 2mv \\$
$\implies \Delta P = 2 \times 0.15 \times 12 \\$
$\therefore \Delta P = 3.6\,{\text{Ns}} \\$
Hence, the impulse imparted to the ball \[3.6\,{\text{Ns}}\] .

So, the correct answer is “Option C”.

Note:
This problem is based on Newton's second law of motion. Students make the mistake mostly in the part while calculating the change in momentum. They fail to understand that the direction of the momentum will get reversed as the velocity is a vector quantity and the direction of the ball leads to the change.