What is the Speed u of the Object at the Height of (1/2)hmax?

Picture tossing a ball straight up into the sky. It goes up, slows down, and then starts to come back down. Let’s find out how fast the ball is going when it’s at half of its highest point.

1. Maximum Height

When the ball reaches its highest point, it’s not moving up or down. All of its initial speed has turned into height. Let’s call this highest point hmaxh_{\text{max}}hmax​.

At the top, the ball’s speed is zero. All the energy has turned into height.

2. Speed at Halfway Up

Now, let’s find out how fast the ball is going when it’s halfway up, which is at a height of 12hmax\frac{1}{2}h_{\text{max}}21​hmax​.

At halfway up, the ball still has some speed and also some height. We can use the idea of energy to figure this out.

Energy Idea:

• Initial Energy: When you first throw the ball, it has energy from its speed. This energy is used to reach the highest point.
• Energy at Halfway Up: At halfway up, the ball has some of its original speed energy left and also some height energy.

The total energy of the ball doesn’t change; it stays the same from when you threw it to when it’s halfway up.

To find the speed at halfway up:

1. At the highest point, all the energy is in height.
2. At halfway up, part of the energy is still in height and the rest is in speed.
   
   

By using the energy idea, we find that the speed of the ball at halfway up is the same as the speed the ball would have had if it were falling from the top of its path to halfway up.

The speed of the ball at half of its highest point is:

Speed=g×hmax\text{Speed} = \sqrt{g \times h_{\text{max}}}Speed=g×hmax​​

Here, ggg is the acceleration due to gravity (which is about 9.8 meters per second squared on Earth), and hmaxh_{\text{max}}hmax​ is the maximum height of the ball.