Question

I throw a ball upward with a speed v. How many times higher would the ball go if instead it were thrown with speed 2v?

a) The ball would reach the same height.
b) The ball would go twice as high.
c) The ball would go four times as high.
d) The ball would go half as high.

Answer 1

Throwing a ball with speed 2v rather than v will make the ball reach a height that is four times greater because the kinetic energy provided to the ball, which converts to potential energy at the maximum height, is four times larger.

Step-by-step explanation:

If you throw a ball upward with speed v, and instead throw it with speed 2v, to determine how many times higher the ball would go, you have to consider the kinetic energy provided to the ball.

The kinetic energy (KE) is given by the formula KE = 1/2 m v^2, where m is the mass and v is the velocity.

When thrown with speed v, the kinetic energy is proportional to v^2. When the speed is doubled to 2v, the new kinetic energy becomes 1/2 m (2v)^2 = 4(1/2 m v^2), which is four times greater than the original kinetic energy.

Since energy due to gravity is conservative, and since the potential energy at the maximum height reached by the ball (PE = mgh, where h is the height) is equal to its initial kinetic energy, then if the kinetic energy is quadrupled, the height reached will also be quadrupled. Therefore, the answer is c) The ball would go four times as high.