Question

A player stops a football weighing $0.5\,kg$ which comes flying towards him with a velocity of $10\,m\,{\sec ^{ - 1}}$ .If the impact lasts for $\dfrac{1}{{50}}th$ second and the ball bounces back with a velocity of $15\,m\,{\sec ^{ - 1}}$ then the average force involved is:
A. $250N$
B. $1250N$
C. $500N$
D. $625N$

Answer 1

In order to solve this question, we will use the concept of Newton’s second law of motion which states that force applied on a body is always equals to the rate of change of momentum of the body, where momentum is defined as the product of mass and velocity of the body.

Formula used:
Momentum of body is calculated as,
$p = mv$
where $m$ is mass and $v$ is the velocity of the body.
Force due the Newton’s second law of motion is written as,
$\left| F \right| = \dfrac{{\left| {dp} \right|}}{{dt}}$

Complete step by step answer:
According to the parameters given in the question, we have mass of the football as $m = 0.5kg$. Let Initial velocity of the ball is denoted by $u = 10\,m\,{\sec ^{ - 1}}$. Now, after bouncing back the final velocity of the ball will be in opposite direction to that of initial velocity direction hence, final velocity will be taken as negative
$v = - 15\,m\,{\sec ^{ - 1}}$

Let initial momentum denoted as ${p_{initial}} = mu$ so we have,
${p_{initial}} = 0.5 \times 10$
$\Rightarrow {p_{initial}} = 5\,kg\,m\,{\sec ^{ - 1}} \to (i)$
Let final momentum denoted as ${p_{final}} = mv$ so we have,
${p_{final}} = - 0.5 \times 15$
$\Rightarrow {p_{final}} = - 7.5\,kg\,m\,{\sec ^{ - 1}} \to (ii)$

Now, let change in momentum is $dp = {p_{final}} - {p_{initial}}$
From equations $(i)and(ii)$ we have,
$dp = - 7.5 - 5$
$\Rightarrow dp = - 12.5\,kg\,m\,{\sec ^{ - 1}}$
Since the magnitude of impact is given as $dt = \dfrac{1}{{50}}\sec$
Now, using the formula of force as $\left| F \right| = \dfrac{{\left| {dp} \right|}}{{dt}}$ we have,
$\left| F \right| = 12.5 \times 50$
$\therefore \left| F \right| = 625\,N$

Hence, the correct option is D.

Note: It should be remembered that, while solving such questions keep the direction of velocity in mind, every time a body reflects or bounce back the direction of velocity gets reversed and also Force and momentum both are a vector quantity and in simpler terms force on a body is just the product of mass and acceleration of the body written as $\vec F = m\vec a$.