Part 1
You are a pirate working for Dread Pirate Roberts. You are in charge of a cannon that exerts a force 10000 N on a cannon ball while the ball is in the barrel of the cannon. The length of the cannon barrel is 2.47 m and the cannon is aimed at a 42degrees angle from the
ground. The acceleration of gravity is 9.8 m/s2 .
If Dread Pirate Roberts tells you he wants the ball to leave the cannon with speed v0 = 76 m/s, what mass cannon ball must you use? Answer in units of kg.
Part 2
Assuming the Dread Pirate Roberts never
misses, how far from the end of the cannon is the ship that you are trying to hit (Neglect dimensions of cannon)?
Answer in units of m.
Part 1: To determine the mass of the cannonball, we can use the equation relating force, mass, and acceleration:
F = m * a
In this case, the force exerted on the cannonball is 10000 N. The acceleration is determined by the angle at which the cannon is aimed and the force of gravity. We can decompose the force into two components, one parallel to the ground and one perpendicular to the ground.
The vertical component of the force is given by:
F_vert = F * sin(theta)
= 10000 N * sin(42 degrees)
≈ 6698 N
This vertical force is responsible for the vertical acceleration of the cannonball, due to the force of gravity. The net vertical force is:
F_net_vert = m * g
= m * 9.8 m/s^2
Setting this equal to the vertical component of the force, we can solve for the mass of the cannonball:
m = F_net_vert / g
= 6698 N / 9.8 m/s^2
≈ 683.67 kg
Therefore, the mass of the cannonball must be approximately 683.67 kg.
Part 2: To determine how far from the end of the cannon the ship is, we can use the range formula for projectile motion:
R = (v0^2 * sin(2*theta)) / g
In this case, the initial velocity of the cannonball is 76 m/s, and the angle at which the cannon is aimed is 42 degrees. The acceleration due to gravity is 9.8 m/s^2. Plugging in the values, we can calculate the range:
R = (76 m/s)^2 * sin(2 * 42 degrees) / 9.8 m/s^2
≈ 414.13 m
Therefore, the ship is approximately 414.13 meters away from the end of the cannon.