Question

If you toss a ball upward with a certain initial speed, it falls freely and reaches a maximum height h. By what factor must you increase the initial speed of the ball for it to reach a maximum height 3h?

Answer 1

`√(3)`, assuming that the air resistance (drag) on the ball is negligible.

Step-by-step explanation:

Let
`g` denote the gravitational field strength (
`g \approx 9.8\; {\rm m\cdot s^(-2)}` near the surface of the earth.) If the drag on the ball is negligible, the ball will be constantly accelerating downward at
`g` while in the air.

Let
`u` denote the initial velocity of this ball. In this example,
`u\!` will be the velocity at which the ball is tossed upwards.

When the ball is at maximum height, its velocity will be
`0`. Let
`v` denote this velocity.

Let
`x` denote the displacement of this ball when it reached maximum height relative to when the ball was initially tossed upward. In this example,
`x = h`.

Let
`a` denote the acceleration of this ball. Under the assumptions,
`a = g \approx (-9.8)\; {\rm m\cdot s^(-2)}`.

The SUVAT equation
`v^(2) - u^(2) = 2\, a\, x` relates these quantities.

Note that since
`v = 0`,
`x = h`, and
`a = (-g)`, this equation becomes:

`0^(2) -u^(2) = 2\, (-g)\, h`.

`u^(2) = 2\, g\, h`.

`u = √(2\, g\, h) = (√(2\, g))\, (√(h))`.

Therefore, replacing
`h` with
`3\, h` will only increase the initial velocity of the ball
`u` by a factor of
`√(3)`.