Question

A machine gun fires 20 bullets per second into a target each bullet weights 150 gms and has a speed of 800 $m{\sec ^{ - 1}}$. the force necessary to hold the gun
A. 800N
B. 1000N
C. 1200N
D. 2400N

Answer 1

Hint:
- Isaac Newton’s second law of motion states that the time rate of change of momentum is equal to the force acting on the particle.
- Momentum is the product of the mass of a particle and its velocity. Momentum is a vector quantity; i.e., it has both magnitude and direction.

Complete step by step answer:
From Newton’s second law it follows that, if a constant force acts on a particle for a given time, the product of force and the time interval (impulse) is equal to the change in the momentum. Inversely, the momentum of a particle is a measure of the time required for a constant force to bring it to rest.
Now according to Newton's second law: the rate of change of momentum with respect to time gives us the force
i.e. \[F = \dfrac{{dp}}{{dt}}\]
now if we put time equal to 1 sec then
$F = dp$(force will be equal to change in momentum)

Now in this question, it is given that 20 bullets are fired from the gun in one second
Mass of each bullet is 150gm and speed is 800$m{\sec ^{ - 1}}$
Now as we know momentum(p)$ = m \times v$
Here m=150gm and velocity of each bullet is =800$m{\sec ^{ - 1}}$
But the mass has to be in kg so we will convert the mass so, mass in kg will be $ = 150 \times {10^{ - 3}}$kg
Substituting the values we get
The momentum of a single bullet will be
$p = 150 \times {10^{ - 3}} \times 800$
$ \Rightarrow p = 120kgm{\sec ^{ - 1}}$
Now this is the mass of one bullet and 20 bullets are fired in one second

So, total momentum will be ${P_t} = 120 \times 20 = 2400kgm{\sec ^{ - 1}}$
Initially, all the bullets were at rest so their initial velocity would be zero hence initio=al momentum will be zero
Now change in momentum will be the final momentum only as the initial momentum was zero
$\therefore dp = 2400kgm{\sec ^{ - 1}}$
Now as we discussed earlier if change in time is 1 second then
$F = dp$
So here 20 bullets are fired in 1 second so automatically change in time becomes 1 sec so
Force will be equal to change in momentum so
$F=2400N$

Hence, the correct answer is option (D).

Note:
- The momentum of the body is also defined as the motion content of the body.
- The momentum of a body depends upon the mass of the body and the velocity of the body.
- The momentum can also be defined as the amount of constant force required to bring a body to rest.