To calculate the power developed by the machine gun, we need to determine how much kinetic energy is imparted to the bullets per second. Power is defined as the rate at which energy is transferred or converted. Here are the steps to find the solution:
### Step 1: Convert the mass of each bullet from grams to kilograms
[tex]\[ \text{Mass of each bullet} = 40 \, \text{grams} \][/tex]
[tex]\[ 1 \, \text{gram} = 0.001 \, \text{kilograms} \][/tex]
[tex]\[ \text{Mass of each bullet} = 40 \, \text{grams} \times 0.001 \, \frac{\text{kilograms}}{\text{gram}} = 0.04 \, \text{kilograms} \][/tex]
### Step 2: Convert the firing rate from bullets per minute to bullets per second
[tex]\[ \text{Firing rate} = 1000 \, \text{bullets per minute} \][/tex]
[tex]\[ 1 \, \text{minute} = 60 \, \text{seconds} \][/tex]
[tex]\[ \text{Firing rate} = \frac{1000 \, \text{bullets}}{60 \, \text{seconds}} = \frac{1000}{60} \, \text{bullets per second} = 16.67 \, \text{bullets per second} \][/tex]
### Step 3: Calculate the kinetic energy of a single bullet
The formula for kinetic energy ([tex]\(KE\)[/tex]) is:
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
where:
- [tex]\(m\)[/tex] is the mass of the bullet,
- [tex]\(v\)[/tex] is the velocity of the bullet.
Given:
[tex]\[ m = 0.04 \, \text{kilograms} \][/tex]
[tex]\[ v = 200 \, \text{meters per second} \][/tex]
[tex]\[ KE = \frac{1}{2} \times 0.04 \, \text{kg} \times (200 \, \text{m/s})^2 \][/tex]
[tex]\[ KE = 0.02 \, \text{kg} \times 40000 \, \text{m}^2/\text{s}^2 \][/tex]
[tex]\[ KE = 800 \, \text{joules} \][/tex]
### Step 4: Calculate the total kinetic energy per second
Since 16.67 bullets are fired per second, the total kinetic energy per second is:
[tex]\[ \text{Total kinetic energy per second} = 800 \, \text{joules} \times 16.67 \, \text{bullets per second} \][/tex]
[tex]\[ \text{Total kinetic energy per second} = 13336 \, \text{joules per second} \][/tex]
### Step 5: Determine the power developed by the machine gun
Power is the rate of energy transfer, so the power developed by the machine gun is:
[tex]\[ \text{Power} = \text{Total kinetic energy per second} = 13336 \, \text{watts} \][/tex]
Thus, the power developed by the machine gun is 13336 watts or 13.336 kilowatts.
