A lift starts ascending with an acceleration of `4 ft//s^(2)`. At the same time a bolt falls from its cieling 6ft above the floor. Find the time taken by it to reach the floor. `g= 32 ft//s^(2)`

Options

A. `(1)/(sqrt(3))s`
B. `(1)/(3)s`
C. `(1)/(sqrt(5))s`
D. `(1)/(5)s`

Answer

To solve the problem, we will analyze the motion of the bolt relative to the lift. Here's a step-by-step solution:

### Step 1: Understand the scenario
The lift is ascending with an acceleration of \(4 \, \text{ft/s}^2\) and the bolt is falling from a height of \(6 \, \text{ft}\) above the floor of the lift. We need to find the time it takes for the bolt to reach the floor of the lift.

### Step 2: Define the relative motion
From the perspective of an observer inside the lift, the bolt is falling while the lift is moving upwards. We will consider the relative motion of the bolt with respect to the lift.

### Step 3: Calculate the relative distance
The distance the bolt needs to cover to reach the floor of the lift is \(6 \, \text{ft}\).

### Step 4: Determine the relative acceleration
The bolt is falling under the influence of gravity, which gives it an acceleration of \(g = 32 \, \text{ft/s}^2\) downward. Since the lift is accelerating upwards at \(4 \, \text{ft/s}^2\), the effective acceleration of the bolt relative to the lift is:
\[ a_{\text{relative}} = g + a_{\text{lift}} = 32 \, \text{ft/s}^2 + 4 \, \text{ft/s}^2 = 36 \, \text{ft/s}^2 \]

### Step 5: Use the equation of motion
We can use the second equation of motion to find the time taken for the bolt to reach the floor of the lift:
\[ S = ut + \frac{1}{2} a t^2 \]
Where:
- \(S = 6 \, \text{ft}\) (the distance to the floor)
- \(u = 0 \, \text{ft/s}\) (initial velocity of the bolt)
- \(a = 36 \, \text{ft/s}^2\) (relative acceleration)

Substituting the values, we have:
\[ 6 = 0 \cdot t + \frac{1}{2} \cdot 36 \cdot t^2 \]
This simplifies to:
\[ 6 = 18 t^2 \]

### Step 6: Solve for time \(t\)
Rearranging the equation gives:
\[ t^2 = \frac{6}{18} = \frac{1}{3} \]
Taking the square root:
\[ t = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}} \, \text{seconds} \]

### Final Answer
The time taken by the bolt to reach the floor of the lift is:
\[ t = \frac{1}{\sqrt{3}} \, \text{seconds} \]