The problem states that a ball is dropped from a height h and it can only attain a height of 4h/5 after bouncing off the floor. We need to find the time it takes for the ball to come to rest when dropped from a height of 1 meter.


To solve this problem, we need to apply the concept of conservation of mechanical energy and the concept of time of flight.


When the ball is dropped from a height h, it will initially have potential energy (mgh) and no kinetic energy. As it falls, this potential energy gets converted into kinetic energy. When the ball reaches the floor, it will have maximum kinetic energy and no potential energy.

After bouncing off the floor, the ball will again rise to a certain height. At this point, it will have potential energy and no kinetic energy. As it rises, this potential energy gets converted into kinetic energy. When the ball reaches the maximum height after bouncing, it will have maximum potential energy and no kinetic energy.

Since the ball is not losing any energy due to air resistance or the finite radius of the ball, the total mechanical energy of the ball is conserved throughout its motion.


The time of flight is the total time taken by the ball to reach the maximum height after bouncing back. It can be calculated using the equation:

t = 2 * sqrt(2h/g)

where t is the time of flight, h is the initial height, and g is the acceleration due to gravity.


Given that the ball can only attain a height of 4h/5 after bouncing, we can set up the following equation:

4h/5 = h - (1/2)gt^2

Solving this equation for t, we can find the time it takes for the ball to reach the maximum height after bouncing. We can then double this time to get the total time of flight.

Let's calculate the value of t:

4h/5 = h - (1/2)gt^2

4h/5 - h = -(1/2)gt^2

(-h/5) = -(1/2)gt^2

h/5 = (1/2)gt^2

2h/5g = t^2

t = sqrt(2h/5g)

Substituting the value of h = 1m and g = 9.8 m/s^2, we can calculate the value of t:

t = sqrt(2*1/5*9.8)

t ≈ 1.97 s

The total time of flight is twice the time it takes for the ball to reach the maximum height after bouncing:

Total time of flight = 2 * t ≈ 2 * 1.97 ≈ 3.94 s

Therefore, the ball will take approximately 3.94 seconds to come to rest when dropped from a height of 1 meter.


The time it will take for the ball to come to rest is approximately 3.94 seconds. Therefore, the correct option is (b) 3.8 s.