You are a pirate working for Dread Pirate Roberts. You are in charge of a cannon that exerts a force 20000 N on a cannon ball while the ball is in the barrel of the cannon. The length of the cannon barrel is 1.88 m and the cannon is aimed at a 44◦ 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 = 83 m/s, what mass cannon ball must you use?
Answer in units of kg.

To find the mass of the cannonball, we can use the equation of motion that relates force, mass, and acceleration.

First, let's analyze the components of the motion of the cannonball. We know that the cannon exerts a force of 20000 N on the cannonball, which creates an acceleration on the cannonball. In this case, the only force acting on the cannonball is the force exerted by the cannon itself.

Once the cannonball leaves the barrel, the only force acting on it is gravity, which causes the cannonball to follow a parabolic trajectory.

To find the mass of the cannonball, we need to determine the acceleration. We can decompose the weight of the cannonball into two components: the vertical component and the horizontal component.

The vertical component of the force is given by:

F_vertical = m * g * cos(θ)

where m is the mass of the cannonball, g is the acceleration due to gravity (9.8 m/s²), and θ is the angle of the cannon with respect to the ground (44°).

Similarly, the horizontal component of the force is given by:

F_horizontal = m * g * sin(θ)

We know that the force exerted by the cannon is equal to the mass of the cannonball times the acceleration:

F_cannon = m * a

where a is the acceleration of the cannonball.

Since we want to find the mass of the cannonball, we can equate the vertical component of the force to the force of the cannon:

m * g * cos(θ) = F_cannon

Simplifying the equation gives:

m = F_cannon / (g * cos(θ))

Now we can plug in the given values:

F_cannon = 20000 N
g = 9.8 m/s²
θ = 44°

Using these values, we can calculate the mass of the cannonball:

m = 20000 N / (9.8 m/s² * cos(44°))

Using a calculator, we find:

m ≈ 270.27 kg

Therefore, the mass of the cannonball must be approximately 270.27 kg.