How many times is it necessary to increase the initial speed of a body thrown vertically upward from the surface of the earth

How many times is it necessary to increase the initial speed of a body thrown vertically upward from the surface of the earth in order to increase its maximum flight height by 4 times?

The coordinates of a body moving with acceleration along the x axis are given by the equation

x = x0 + vt + at ^ 2/2.

Let’s point the x-axis up. The origin is at the throwing point, that is, x0 = 0. The acceleration g is directed to the origin, and the initial velocity is v1:

x = v1t – gt ^ 2/2 (1).

Let us find the speed as the derivative with respect to t of x:

x ‘= v1 – 2gt / 2 = v1 – gt.

The height x will be maximum at the moment t1, at x ‘= 0:

0 = v1 – gt1;

v1 = gt;

t1 = v1 / g (2).

Substitute (2) into (1):

x1 = (v1 * v1) / g – g * (v12 / g2) / 2 = v1 ^ 2 / 2g.

v1 ^ 2 = 2x1g.

At a certain speed v2, the lifting height should be 4 times higher:

4×1 = v2 ^ 2 / 2g;

v2 ^ 2 = 8x1g;

v2 ^ 2 / v1 ^ 2 = (8x1g) / (2x1g) = 4.

v2 / v1 = 2.

v2 = 2v1.

Answer. The speed should be doubled.