Three identical balls are thrown from a cliff of height, h, above a horizontal plane. One ball is thrown at an angle of 45 degrees above the horizon, the second is thrown horizontally, and the third is thrown at an angle of 30 degrees below the horizon (so, part of this velocity is downward). If all the balls are given the same initial speed, which one will have the greatest speed at the moment just before it hits the ground?

The work theorem says...

final KE=initial KE+changeinPE

Does angle enter into that?

To determine which ball will have the greatest speed just before hitting the ground, we need to consider the motion of each ball separately.

Let's analyze the motion of each ball:

1. Ball thrown at an angle of 45 degrees above the horizon:
When the ball is thrown at an angle of 45 degrees above the horizon, it will have a horizontal velocity component and a vertical velocity component. The vertical component will cause the ball to rise in the air, reaching a maximum height, and then fall back down to the ground. The horizontal velocity component remains constant throughout the entire trajectory. So, the ball will travel the farthest horizontally before hitting the ground.

2. Ball thrown horizontally:
When the ball is thrown horizontally, all of its initial velocity is in the horizontal direction, and it doesn't have any vertical velocity component. Since there is no vertical force acting on the ball, it will fall downward in a straight line without any horizontal acceleration. This means that the ball will have a constant horizontal velocity. As a result, it will have the same horizontal distance traveled as the first ball, but it will take less time to reach the ground.

3. Ball thrown at an angle of 30 degrees below the horizon:
When the ball is thrown at an angle of 30 degrees below the horizon, it will have a horizontal velocity component and a downward vertical velocity component. Similar to the first ball, the vertical component will cause the ball to rise to a certain peak height and then fall back down. However, due to the downward velocity component, the ball will reach its maximum height at a lower point compared to the first ball. Thus, it will have a shorter horizontal distance traveled before reaching the ground.

In conclusion, the ball thrown horizontally will have the greatest speed just before hitting the ground. While the ball thrown at an angle of 45 degrees will have the same horizontal distance traveled, it will take a longer time to reach the ground due to its parabolic trajectory. The ball thrown at an angle of 30 degrees below the horizon will have a shorter horizontal distance traveled and also a shorter time of flight.