A rubber ball is bounced on the floor of a room which has its ceiling at a height of 3.2 m from the floor. The ball hits the floor with a speed of 10 m`//`s, and rebounds vertically up. If all collisions simply reverse the velocity of the ball, without changing its speed, then how long does it take the ball for a round trip, from the moment it bounces from the floor to the moment it returns back to it ? Acceleration due to gravity is 10 `m//s^2`

Options

A. 4 s
B. 2 s
C. 0.8 s
D. 1.2 s

Explanation

To solve the problem of how long it takes for the rubber ball to complete a round trip (from the moment it bounces off the floor to the moment it returns back to the floor), we can break the solution down into several steps.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Height of the ceiling from the floor, \( h = 3.2 \, \text{m} \)
- Speed of the ball when it hits the floor, \( v = 10 \, \text{m/s} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

2. **Calculate the Time to Reach the Ceiling:**
- When the ball rebounds, it travels upwards to the ceiling. We can use the first equation of motion:
\[ v^2 = u^2 + 2as \]
where:
- \( v \) = final velocity at the ceiling (0 m/s, since it stops momentarily at the peak),
- \( u \) = initial velocity (10 m/s, the speed at which it rebounds),
- \( a \) = acceleration (-10 m/s², since gravity acts downwards),
- \( s \) = distance (3.2 m, the height to the ceiling).

Rearranging the equation gives:
\[ 0 = (10)^2 + 2(-10)(s) \]
\[ 0 = 100 - 20s \]
\[ 20s = 100 \implies s = 5 \, \text{m} \]
Since the height to the ceiling is 3.2 m, we can find the time taken to reach the ceiling using:
\[ v = u + at \implies 0 = 10 - 10t \implies t = 1 \, \text{s} \]

3. **Calculate the Time to Fall Back to the Floor:**
- The time taken to fall back down from the ceiling to the floor will be the same as the time taken to rise to the ceiling because of symmetry in motion (ignoring air resistance).
- Therefore, the time to fall back down, \( t_{\text{down}} = t_{\text{up}} = 1 \, \text{s} \).

4. **Calculate the Total Time for the Round Trip:**
- The total time for the round trip is the sum of the time to go up and the time to come down:
\[ t_{\text{total}} = t_{\text{up}} + t_{\text{down}} = 1 \, \text{s} + 1 \, \text{s} = 2 \, \text{s} \]

### Final Answer:
The total time taken for the ball to complete a round trip is **2 seconds**.

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