A 8.00 multiplied by 107 kg battleship originally at rest fires a 1400 kg artillery shell horizontally with a velocity of +550 m/s.
(a) If the shell is fired straight aft, there will be negligible friction opposing the ships recoil velocity. Calculate its recoil velocity.
(b) Calculate the increase in internal kinetic energy (that is, for the ship and the shell). This is less than the energy released by the gun powder -- significant heat is generated.
To calculate the recoil velocity of the battleship, we can use the principle of conservation of momentum. According to this principle, the total momentum before firing the shell is equal to the total momentum after firing the shell.
(a) Calculation of the recoil velocity of the battleship:
The equation for momentum is:
Momentum = mass × velocity
For the battleship:
Initial momentum = mass × velocity = (8.00 kg) × (0 m/s) = 0 kg·m/s
For the artillery shell:
Final momentum = mass × velocity = (1400 kg) × (550 m/s) = 770,000 kg·m/s
According to the conservation of momentum principle:
Total initial momentum = Total final momentum
Therefore, the recoil velocity of the battleship can be calculated as follows:
Recoil velocity = Total final momentum of the system / Mass of the battleship
Recoil velocity = 770,000 kg·m/s / 8.00 kg = 96,250 m/s (approx.)
So, the recoil velocity of the battleship is approximately 96,250 m/s.
(b) Calculation of the increase in internal kinetic energy:
To calculate the increase in internal kinetic energy, we need to add up the kinetic energies of the battleship and the artillery shell before and after the firing.
The equation for kinetic energy is:
Kinetic energy = (1/2) × mass × velocity^2
For the battleship:
Initial kinetic energy of the battleship = (1/2) × (8.00 kg) × (0 m/s)^2 = 0 J
For the artillery shell:
Initial kinetic energy of the artillery shell = (1/2) × (1400 kg) × (0 m/s)^2 = 0 J
After firing, the velocities become:
Velocity of the battleship = recoil velocity = 96,250 m/s
Velocity of the artillery shell = 550 m/s
Final kinetic energy of the battleship = (1/2) × (8.00 kg) × (96,250 m/s)^2
Final kinetic energy of the artillery shell = (1/2) × (1400 kg) × (550 m/s)^2
The increase in internal kinetic energy is the difference:
Increase in internal kinetic energy = (Final kinetic energy of the battleship + Final kinetic energy of the artillery shell) - (Initial kinetic energy of the battleship + Initial kinetic energy of the artillery shell)
Therefore, you can calculate the increase in internal kinetic energy by substituting the values into the equation.
Note: The actual energy released by gunpowder may generate additional heat, so the increase in internal kinetic energy, as calculated here, represents the kinetic energy of the system.