This relationship can be seen from the following equation:
$$v_f^2 = v_i^2 + 2ad$$
where:
- \(v_f\) is the final velocity of the object (which is 0 m/s when it reaches its maximum height)
- \(v_i\) is the initial velocity of the object
- \(a\) is the acceleration due to gravity (-9.8 m/s^2)
- \(d\) is the displacement of the object (which is the maximum height it reaches)
Solving this equation for \(d\), we get:
$$d = \frac{v_i^2}{2a}$$
This equation shows that the maximum height reached by an object is proportional to the square of its initial velocity. In other words, if you double the initial velocity, the object will reach four times the height.
This relationship can be seen in the following table:
| Initial Velocity (m/s) | Maximum Height Reached (m) |
|---|---|
| 10 | 5 |
| 20 | 20 |
| 30 | 45 |
| 40 | 80 |
| 50 | 125 |
As you can see, the maximum height reached by an object increases dramatically as the initial velocity increases.
