Aball is thrown at some angle. the ball is in the air for 4.5 seconds before it hits. if it travels 45 meters before it hits the ground, what was the initial velocity of the ball? (magnitude and direction)

Assume the ball was thrown at ground level, so its initial position vector is . Let denote the ball's initial speed and the angle at which the ball is thrown.

Over time, its position changes according to , where the components are determined by

where is the acceleration due to gravity. We know the ball is in the air for 4.5 seconds and that it travels a horizontal distance of 45 meters, which means

Divide the second equation by the first to eliminate , and we can solve for , expecting some solution within :

Then plugging this into either equation above will tell us the ball's initial speed: