Question

A hockey ball of mass $200\;g$ travelling at $10\;m/s$ is struck by a hockey stick so as to return it along its original path with a velocity at $5\;m/s$. Calculate the change of linear momentum occurred in the motion of the hockey ball by the force applied by the hockey stick.

Answer 1

The change in velocities of the hockey ball in motion and its respective directions after collision must be considered since velocity is a vector quantity. The formula for momentum is applied in order to compute the change in linear momentum. The concept of conservation of linear momentum and Newton's second law of motion in terms of force and momentum is also applied.

Formula used:
The formula for the change in linear momentum is given by:
$p = m\Delta v$
Where, $m$ is the mass of the body and $\Delta v$ is the change in velocity of the body.

Complete answer:
First, let us look into the concept of linear momentum. Momentum is defined as the measure of the quantity of motion in a body. Momentum comes into play when a body is said to be in motion and is said to have a certain velocity as well as some mass associated with it. The momentum is equal to the product of the mass of the body and the velocity of the body.

Hence it is mathematically given by the equation:
$p = mv$ --------($1$)
Newton's Second law of motion states that the rate of change of momentum of a body is directly proportional to the force applied on it and the change in momentum occurs in the direction of the applied force. The question mentions that there is a change in the velocity of the hockey ball after it is struck. This is because when a body is in motion it continues to be in motion until an external force is applied in it as per Newton’s first law of inertia.

This means that when struck by the hockey stick, due to the external force applied the ball will start to move in the direction of the applied force. This reverses its direction as the ball starts to move in the direction opposite to the direction which it was travelling in.Hence by this concept, equation ($1$) becomes:
$p = m\Delta v$ --------($2$)
The change in velocity is due to the external force applied by the hockey stick and as per the concept of linear momentum it is said that the body starts to move with a different velocity when it collides another object or when there is a force applied on it.

Since velocity is a vector quantity the direction in which the body moves with a certain velocity is also required to be considered and this can be indicated by a change in its sign signifying the change in its direction.Knowing this, let us now extract the given data from the question. The mass of the hockey ball as well as the initial and final velocities of the ball are also given. We are asked to find out the change in the linear momentum.

Given, $m = 200\;g$
$\Rightarrow {v_1} = 10\;m/s$
$\Rightarrow {v_2} = - 5\;m/s$
Here, ${v_2}$ value has a negative sign indicating that the direction of the hockey ball after the hockey stick strikes it moves in a direction opposite to the direction it was initially moving in. Hence the direction of ${v_2}$ is opposite with respect to ${v_1}$. We now convert the mass in grams into kilograms since the SI unit of mass is kg.
Since, $1kg = 1000g$
By unitary method,
$200\;g = \dfrac{{200}}{{1000}}kg$
$\Rightarrow 200\;g= 0.2kg$

Hence, $m = 0.2kg$
Equation ($2$) can also be written as:
$p = m\left( {{v_2} - {v_1}} \right)$ ------($3$)
The mass of the ball is constant and does not change after getting struck by the hockey stick and we consider the same hockey ball even after collision.
By substituting the given values inn equation ($3$) we get:
$p = 0.2( - 5 - 10)$
We now solve out the expression to get the value for the change in momentum.
$p = 0.2( - 15)$
$\therefore p = - 3\;Ns$

Hence the change in linear momentum is $- 3\;Ns$.

Note: A common mistake made in this problem is that the direction of the initial and final velocities are often neglected which is wrong. The corresponding signs of the velocities must also be considered. An alternative way to solve the above problem is by finding out the initial and final momentums of the ball separately, before and after collision respectively and the change in the momentum is given by its difference.