# Momentum in a Many-Particle System

1. Oct 31, 2007

### Jacobpm64

1. The problem statement, all variables and given/known data
A Maxim machine gun fires 450 bullets per minute. Each bullet has a mass of 14 g and a velocity of 630 m/s.
(a) What is the average force that the impact of these bullets exerts on a target? Assume the bullets penetrate the target and remain embedded in it.
(b) What is the average rate at which the bullet deliver their energy to the target?

2. Relevant equations
We probably need the fact that $$\frac{dp}{dt} = F_{ext}$$. (i'm not sure if the system is isolated).

3. The attempt at a solution
(a)
Well, momentum = mass * velocity
So, I can find the momentum of one bullet easily:
$$p = mv$$
$$p = (0.014 kg) * (630 \frac{m}{s})$$
$$p = 8.82 kg * \frac{m}{s}$$ <--- momentum of a single bullet.

Now, I can set up the equation:
$$p_{total} = 450 (0.014) (630) t$$ <--- because there's 450 bullets.. and I put the t in there because you don't know how many minutes.

Now, you want force, and since $$\frac{dp}{dt} = F_{ext}$$, you can differentiate the above expression:
$$\frac{dp}{dt} = 450 (0.014) (630)$$
$$F_{ext} = 3969 N$$

(b)
Well, the kinetic energy of the bullets is $$K = 450 * \frac{1}{2} * m * v^2 * t$$
With numbers, $$K = 450 * \frac{1}{2} * (0.014) * (630)^2 * t$$

Now to get the rate at which the bullets deliver their energy (rate of change of energy), we differentiate this expression:
$$\frac{dK}{dt} = 450 * \frac{1}{2} * 0.014 * (630)^2$$
$$\frac{dK}{dt} = 1250235 J / min$$

Is this correct?