Question

A cannon is at a castle on a cliff and at a hight of 200m above the sea level. The cannon barrel points at an angle of +30.0◦ from the horizontal (pointing slightly upwards) and can give a maximum speed to the cannon ball of 70.0 m/s. Ignoring air resistance how close in the horizontal displacement should a ship be in order for the cannon to be able to hit the ship. (hints: identify the motion, write the equations and calculate first the total time the ball is in the air). A ship has some size so we do not need an accuracy better than 10 m.

Answer 1

To find the minimum distance that the ship must be in order for the cannonball to be able to hit the ship, the initial velocity of the the cannon must be maximum. Given the maximum speed of the cannonball is 70.0m/s, let the cannonball be fired at an initial speed of 70.0m/s.

The horizontal component of the intial velocity is

The vertical component of the initial velocity is

Acceleration along the horizontal direction is zero. The horizontal distance covered by the cannonball is

where T is the total time the ball is in the air.

The vertical acceleration of the ball is equal to the gravitational acceleration. The motion is governed by the following kinematics equation

The ship is at at the sea level h=0, the height of the cannonball when it will the ship is zero. The initial height of the cannonball is equal to the height of the cliff (200m).

Substituting we get

This is a quadratic equation in time T. The roots of a quadratic equation are

Therefore

We are interested in time after the ball was fired, that is we are in interested in the positive time

Substituting into equation (1) we get

So, the ship must be at least 660m away from the cliff in order for the cannonball to hit the ship.