Understanding the Physics

* Gravity: The primary force acting on the ball is gravity, which constantly pulls it downward.

* Initial Velocity: The speed at which you throw the ball upward.

* Maximum Height: The point where the ball momentarily stops moving upward before falling back down.

The Relationship

The maximum height reached by a ball thrown upward is directly related to the square of its initial velocity. Here's why:

1. Energy Conservation: As the ball goes up, its initial kinetic energy (energy of motion) is converted into potential energy (energy due to its height).

2. Kinetic Energy Formula: Kinetic energy (KE) = (1/2) * mass * (velocity)^2

3. Potential Energy Formula: Potential energy (PE) = mass * gravity * height

The Calculation

* Initial Velocity: Let the initial velocity be 'v'.

* Double Initial Velocity: The new initial velocity is '2v'.

Since the maximum height is proportional to the square of the initial velocity:

* Original Height (h): h ∝ v²

* New Height (h'): h' ∝ (2v)² = 4v²

Conclusion

If you double the initial velocity, the maximum height reached by the ball will be four times the original height (h').