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 find the mass of the cannonball, we can use Newton's second law of motion, which states that the force exerted on an object is equal to the mass of the object multiplied by its acceleration.
In this case, we know the force exerted by the cannon on the cannonball is 10000 N. The acceleration of the cannonball can be calculated using the equation:
acceleration = change in velocity / time
Since the cannonball starts from rest, the initial velocity (v0) is 0 m/s, and the final velocity (vf) is 76 m/s. The time taken for the cannonball to reach this velocity can be calculated using the equation:
vf = acceleration * time
Rearranging the equation, we get:
time = vf / acceleration
Using the value of gravity (9.8 m/s^2) as the acceleration, we can calculate the time taken for the cannonball to reach its final velocity:
time = 76 m/s / 9.8 m/s^2
Now that we have the time taken, we can calculate the distance traveled by the cannonball in the barrel of the cannon. This can be calculated using the equation:
distance = initial velocity * time + (1/2) * acceleration * time^2
Substituting the known values, we can solve for the distance:
distance = 0.5 * 9.8 m/s^2 * (76 m/s / 9.8 m/s^2)^2
Now that we have the distance, we can calculate the average force exerted on the cannonball using the equation:
force = mass * acceleration
Since the acceleration is constant, the average force exerted is equal to the force exerted in the barrel of the cannon (10000 N). Rearranging the equation, we can solve for the mass of the cannonball:
mass = force / acceleration
mass = 10000 N / 9.8 m/s^2
The mass of the cannonball must be approximately 1020.41 kg.
Part 2:
To find the distance from the end of the cannon to the ship, we can use trigonometry and the equation for horizontal motion.
The horizontal distance traveled by the cannonball can be calculated using the equation:
distance = initial velocity * time
Since the initial velocity is equal to the horizontal component of the cannonball's initial velocity, which can be calculated as:
initial velocity (horizontal) = initial velocity * cos(angle)
where the angle is 42 degrees, we can substitute this value:
initial velocity (horizontal) = 76 m/s * cos(42 degrees)
Now that we have the horizontal velocity, we can calculate the time taken for the cannonball to reach the ship using the equation:
time = distance / initial velocity (horizontal)
Since the distance we want to find is equal to the distance traveled by the cannonball, we substitute the known values and solve for time:
time = distance / (76 m/s * cos(42 degrees))
Given that the velocity in the horizontal direction remains constant (assuming no air resistance), the distance traveled by the cannonball horizontally is equal to the distance from the end of the cannon to the ship.
Hence, the distance from the end of the cannon to the ship is equal to:
distance = (76 m/s * cos(42 degrees)) * (76 m/s * cos(42 degrees)) / (9.8 m/s^2)
The distance from the end of the cannon to the ship is approximately 591.64 meters.