Question

A mysterious tower has been constructed at the edge of a gridiron football field. Even more mysteriously, the field has been marked carefully in meters instead of yards. On top of the tower is a well calibrate ball launcher that will fire heavy ball at a velocity of 14.7 m/s at an angle of 20 degrees abive the horizontal. The ball lands 46.1 meters away from the base of the tower.

A) How long does it take for the ball to reach max height?

B)How long does it take for the ball to return to the height from which it was launched?

C) What is the total flight time of the ball?

D) How tall is the tower?

Answer 1

Starting with the given information,

Alright. We will start by choosing the appropriate co-ordinate system. Let our origin lie at the top of our tower, such that below the tower will be the negative y-axis. At t = 0 seconds, the ball is shot out of a 'ball launcher', with intial velocity 14.7 m/s at angle of degrees with the x-axis. So, our velocity will have two components, x and y, which are

Keep in mind that the force of gravity will only affect the y-component of the velocity. Now, the ball will follow a trajectory such that it will reach a maximum height while also traveling away from the tower. When it does reach maximum height, its y-component of velocity will be zero.

(A) Keeping that in mind, we will use the kinematic equation of motion,

In our case,

Since here, the horizontal component plays no role. Now, velocity at maximum height will be zero, so,

(B) It takes about a half second for the ball to reach the maximum height. Now. Since there are no extra forces on the system that ruin our perfect equations like air resistance and wind, the time taken to go up is equal to the time taken to come down. So, our ball will come down to the height (y = 0) it started at,

(C) But, our ball doesn't stop there. It keeps going, further down till it hits the ground. When it does hit the ground, it has covered 46.1 meters from the base of the tower. Again using our x-components this time,

This is when our ball hits the ground. It is the total time of flight.

(D) Now, we need to find the height of the tower. Which is an easy enough thing. In our co-ordinate system we have to find the value in the negative y-axis at t = 3.337 seconds.

So, from the ground, our tower is 37.84 meters tall.