A revolutionary cannon, with a mass of 2000kg, fires a 20kg ball horizontally. The cannonball has a speed of 140m/s after it has left the barrel. The cannon carriage is on a flat platform and is free to roll horizontally. What is the speed of the cannon immediately after it was fired? Answer in units of m/s.
Use the law of conservation of momentum. The total momentum of the cannon and the cannonball after firing is zero. Make sure the cannon and cannonball momenta have opposite signs when you solve the equation.
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To calculate the speed of the cannon immediately after it was fired, we need to use the law of conservation of momentum. According to this law, the total momentum before the firing should equal the total momentum after the firing.
The momentum of an object is calculated by multiplying its mass with its velocity. For the cannon, we have a mass of 2000 kg and an unknown velocity after firing. For the cannonball, we have a mass of 20 kg and a velocity of 140 m/s after leaving the barrel.
Let's denote the velocity of the cannon as V_c and the velocity of the cannonball as V_b. Since the cannon carriage is free to roll horizontally, the system is isolated, and the total momentum will be conserved.
Before the firing, both the cannon and cannonball were at rest, so their velocities were zero. Therefore, the initial momentum of the system is zero:
Initial momentum = (mass of cannon x velocity of cannon) + (mass of cannonball x velocity of cannonball)
0 = (2000 kg x 0 m/s) + (20 kg x 0 m/s)
After the firing, the cannonball moves with a velocity of 140 m/s, while the cannon recoils in the opposite direction with an unknown velocity V_c. Therefore, the momentum after firing is:
Final momentum = (mass of cannon x velocity of cannon) + (mass of cannonball x velocity of cannonball)
0 = (2000 kg x V_c) + (20 kg x 140 m/s)
Now we can solve the equation for V_c, the velocity of the cannon immediately after it was fired:
0 = (2000 kg x V_c) + (20 kg x 140 m/s)
Multiplying 20 kg by 140 m/s gives us 2800 kg*m/s. We can subtract this term from both sides:
-2800 kg*m/s = 2000 kg x V_c
Dividing both sides by 2000 kg gives us:
-2800 kg*m/s / 2000 kg = V_c
Simplifying the expression gives us the final result:
V_c = -1.4 m/s
Therefore, the speed of the cannon immediately after it was fired is 1.4 m/s in the opposite direction of the cannonball.