Question

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 1.69 m and the cannon is aimed at a 50◦ angle from the ground. the acceleration of gravity is 9.8 m/s 2 . if dread pirate roberts tells you he wants the ball to leave the cannon with speed v0 = 74 m/s, what mass cannon ball must you use? answer in units of kg.

Answer 1

The work done in the cannon has to be equal to the total energy of the cannonball when it leaves the barrel.

Work done:
$W = \int\limits {\overrightarrow F \cdot} \, \overrightarrow {dx} = F \int\limits \, dx = Fx$
where F is the force on the cannonball and x is the length of the barrel.

The total energy of the cannonball:
$E = \frac{1}{2} mv^2 + mgh$
where m is the mass of the ball, v is the velocity, g is the gravitational acceleration and h is the height above ground of the ball.

Combining both equations, where θ is the angle of the cannon:
$Fx = \frac{1}{2} mv^2 + mgh = \frac{1}{2} mv^2 + mgxsin\theta \\ \\ m = \frac{Fx}{ \frac{1}{2} v^2 + gx sin\theta} = 12,29$