With what initial speed should the ball be thrown down from a height h so that it bounces to a height of 2h?

With what initial speed should the ball be thrown down from a height h so that it bounces to a height of 2h? Consider the impact on the ground as absolutely elastic

Given:

h is the initial height from which the ball was thrown down;

m is the mass of the ball;

g = 9.8 m / s2 – acceleration of gravity (constant value near the earth’s surface);

h1 = 2 * h – required height.

It is required to determine v0 (m / s) – with what initial speed it is necessary to throw the ball so that it bounced to the height h1.

According to the condition of the problem, the impact on the ground will be absolutely elastic. Then, according to the law of conservation of energy, the sum of kinetic and potential energies at height h must be equal to potential energy at height h1, that is:

Epot (h) + Ekin (h) = Epot (h1);

m * g * h + m * v0 ^ 2/2 = m * g * h1;

m * g * h + m * v0 ^ 2/2 = 2 * m * g * h;

g * h + v0 ^ 2/2 = 2 * g * h;

v0 ^ 2/2 = 2 * g * h – g * h;

v0 ^ 2/2 = g * h;

v0 ^ 2 = 2 * g * h;

v0 = (2 * g * h) ^ 0.5 = (2 * 9.8 * h) ^ 0.5 = (19.6 * h) ^ 0.5 = 4.4 * h ^ 0.5.

Answer: The initial speed should be 4.4 * h ^ 0.5.