Consider a gravity free hall in which an experimenter of mass 50 kg is resting on a 5 kg pillow, 8 ft above the floor of the hall. He pushes the pillow down so that it starts falling at a speed of 8 ft/s. The pillow makes a perfectly elastic collision with the floor, rebounds and reaches the experimenter\'s head. Find the time elapsed in the process.

Options

A. 2.2 s
B. 1.1 s
C. 4.4 s
D. 3.3 s

Answer

To solve the problem step by step, we will analyze the motion of the pillow and the experimenter, taking into account the elastic collision and the distances involved.

### Step 1: Understand the Initial Conditions
The experimenter has a mass of 50 kg and is resting on a pillow of mass 5 kg. The pillow is initially at a height of 8 ft above the floor and is pushed down with an initial speed of 8 ft/s.

### Step 2: Calculate the Time Taken for the Pillow to Hit the Floor
The pillow is falling under the influence of gravity-free conditions, so we will consider its initial velocity and the distance it travels.

- **Initial velocity (u)** of the pillow = 8 ft/s (downward)
- **Distance (h)** to the floor = 8 ft

Using the equation of motion:
\[ h = ut + \frac{1}{2} a t^2 \]
Since there is no gravity, the acceleration \( a = 0 \):
\[ 8 = 8t \]
Solving for \( t \):
\[ t_1 = \frac{8}{8} = 1 \text{ second} \]

### Step 3: Determine the Pillow's Rebound Velocity
Since the collision with the floor is perfectly elastic, the pillow will rebound with the same speed it hit the floor, which is 8 ft/s (upward).

### Step 4: Calculate the Height from the Pillow to the Experimenter's Head
After the pillow rebounds, it will travel upward to the experimenter's head. The distance from the pillow to the experimenter's head is the initial height plus the distance the pillow fell:
\[ H = 8 \text{ ft (height to floor)} + 0.8 \text{ ft (height moved by experimenter)} = 8.8 \text{ ft} \]

### Step 5: Calculate the Relative Velocity After the Rebound
The experimenter moves upward with a velocity of 0.8 ft/s (as calculated from the downward motion of the pillow). The relative velocity of the pillow after the rebound with respect to the experimenter is:
\[ v_{\text{relative}} = v_{\text{pillow}} - v_{\text{experimenter}} = 8 \text{ ft/s} - 0.8 \text{ ft/s} = 7.2 \text{ ft/s} \]

### Step 6: Calculate the Time Taken for the Pillow to Reach the Experimenter's Head
Using the distance and the relative velocity, we can find the time taken for the pillow to reach the experimenter's head:
\[ t_2 = \frac{H}{v_{\text{relative}}} = \frac{8.8 \text{ ft}}{7.2 \text{ ft/s}} \]
Calculating \( t_2 \):
\[ t_2 \approx 1.22 \text{ seconds} \]

### Step 7: Calculate the Total Time Elapsed
The total time elapsed is the sum of the time taken for the pillow to hit the floor and the time taken for it to reach the experimenter's head:
\[ t_{\text{total}} = t_1 + t_2 = 1 \text{ second} + 1.22 \text{ seconds} = 2.22 \text{ seconds} \]

### Final Answer
The total time elapsed in the process is **2.22 seconds**.

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