When a ball is dropped from a height of 90 m, it gains potential energy. As it falls towards the ground, this potential energy is converted into kinetic energy of motion. However, at each collision with the floor, the ball loses one-tenth of its speed due to the loss of energy in the form of sound and heat.


When the ball is dropped from a height of 90 m, it has an initial velocity of 0 m/s. This is because the ball is initially at rest. As it falls towards the ground, its velocity increases due to the acceleration due to gravity.


The acceleration due to gravity is a constant value of 9.8 m/s^2. This means that the velocity of the ball increases by 9.8 m/s every second.


When the ball collides with the floor for the first time, it loses one-tenth of its velocity. This means that its velocity decreases by 9.8/10 = 0.98 m/s. The ball then starts to rise again due to the force of the collision.


The maximum height reached by the ball after the first collision can be calculated using the formula:

h = (v^2)/(2g)

where h is the maximum height, v is the velocity just before the collision, and g is the acceleration due to gravity.

Substituting the values, we get:

h = (89.02^2)/(2*9.8) = 398.7 m

This means that the ball rises to a height of 398.7 m after the first collision.


After each collision with the floor, the ball loses one-tenth of its velocity. This means that its velocity decreases by 0.98 m/s after each collision. The ball then rises to a certain height before falling back down again. The time taken for the ball to reach the floor after each collision can be calculated using the formula:

t = (2h)/(v+u)

where t is the time taken, h is the maximum height reached by the ball, v is the final velocity of the ball just before it hits the floor, and u is the initial velocity of the ball just after the collision.

Substituting the values, we get:

t = (2*398.7)/(0+89.02) = 8.96 s

This means that the ball takes 8.96 s to reach the floor after the first collision.


The speed-time graph of the motion of the ball between t = 0 to 12 s can be plotted as follows:

- From t = 0 to t = 4.51 s, the speed of the ball increases linearly from 0 m/s to 89.02 m/s due to the acceleration due to gravity.
- At t = 4.51 s, the ball collides with the floor for the first time and its speed decreases to 88.04 m/s.
- From t = 4.51 s to t = 13.47 s, the ball moves upwards