Question

A machine gun fires a bullet of mass 50gm with velocity 1000m/s. If the average force acting on the gun is 200N then find out the maximum number of bullets fired per minute.

Answer 1

Newton’s second law of motion in terms of momentum states that total external force equals to the ratio of the change in momentum and the time it takes to change, $F = \dfrac{{\Delta P}}{{\Delta t}},P = m \times \Delta v$, where P is the momentum, m is the mass, t is the time taken and v is the velocity. So to calculate the number of bullets fired by the gun use the formula $F = \dfrac{{n\Delta P}}{{\Delta t}}$, where n will be the no. of bullets and F will be the given total Force acting on the gun value.

Complete step by step answer:
We are given that a machine gun fires a bullet of mass 50gm with velocity 1000m/s and the average force acting on the gun is 200N.
We have to calculate the maximum number of bullets fired from the gun per minute.
Force (F) can be defined as $F = \dfrac{{\Delta P}}{{\Delta t}},P = m \times \Delta v$
So when n no. of bullets are fired in a second then the total force will be $F = \dfrac{{n\Delta P}}{{\Delta t}}$, where t is 1 second.
Force is equal to 200N, mass is 30 grams, velocity is 1000m/s.
$F = \dfrac{{n\Delta P}}{{\Delta t}} \\\ P = m \times \Delta v \\\ m = 50gm \\\ 1gm = {10^{ - 3}}kg \\\ 50gm = 50 \times {10^{ - 3}} = 0.05kg \\\ v = 1000m/s \\\ t = 1\sec \\\ F = 200N \\\ 200 = \dfrac{{n \times 0.05 \times 1000}}{1} \\\ n = \dfrac{{200}}{{0.05 \times 1000}} \\\ \implies n = \dfrac{{200}}{{50}} \\\ \therefore n = 4 \\\$
Therefore, the number of bullets fired from the machine gun in a second is 4.
1 minute= 60 seconds.
Number of bullets fired from the machine gun in a minute is $4 \times 60 = 240$
Maximum number of bullets fired from the gun per minute is 240.

Note: Another approach
Force is defined as the product of mass and acceleration.
$F = m \times a$
Acceleration is the ratio of change in velocity and the time taken to change.
$a = \dfrac{{\Delta v}}{{\Delta t}} \\\ F = \dfrac{{m\Delta v}}{{\Delta t}} \\\$
When n number of bullets is fired from the machine gun in a minute then the total force will be $F = \dfrac{{n \times m \times \Delta v}}{{\Delta t}} \\\ F = 200N \\\ m = 50grams = 0.05kg \\\ t = 60seconds \\\ v = 1000m/s \\\ 200 = \dfrac{{n \times 0.05 \times 1000}}{{60}} \\\ n = \dfrac{{200 \times 60}}{{50}} \\\ n = 240 \\\$
Maximum 240 bullets can be fired from the machine gun in a minute.