An elevator is accelerating upward at a rate of `6ft(sec^(2)` when a bolt from its celling falls to the floor of the lift (Distance=9.5feet). The time taken (in seconds) by the falling bolt to hit the floor is (take `g=32ft//sec^(2)`)

Options

A. `sqrt(2)`
B. `(1)/sqrt(2)`
C. `2sqrt(2)`
D. `(1)/(2sqrt(2))`

Explanation

To solve the problem, we need to determine the time taken by a bolt to fall from the ceiling of an accelerating elevator to the floor of the elevator. Here’s the step-by-step solution:

### Step 1: Understand the scenario
The elevator is accelerating upward at a rate of \(6 \, \text{ft/s}^2\). The bolt falls from the ceiling of the elevator to the floor, which is \(9.5 \, \text{feet}\) apart. The acceleration due to gravity \(g\) is given as \(32 \, \text{ft/s}^2\).

### Step 2: Determine the effective acceleration of the bolt
Since the elevator is accelerating upward, the effective acceleration acting on the bolt (with respect to the elevator) will be the sum of the gravitational acceleration and the elevator's acceleration. Therefore, the effective acceleration \(a\) of the bolt is:
\[ a = g + \text{(acceleration of the elevator)} = 32 \, \text{ft/s}^2 + 6 \, \text{ft/s}^2 = 38 \, \text{ft/s}^2 \]

### Step 3: Use the kinematic equation
We can use the kinematic equation for motion under constant acceleration to find the time \(t\) taken for the bolt to fall:
\[ s = ut + \frac{1}{2} a t^2 \]
Where:
- \(s\) is the distance fallen (which is \(9.5 \, \text{feet}\)),
- \(u\) is the initial velocity (which is \(0 \, \text{ft/s}\) since the bolt starts from rest),
- \(a\) is the effective acceleration (\(38 \, \text{ft/s}^2\)),
- \(t\) is the time in seconds.

Substituting the known values into the equation:
\[ 9.5 = 0 + \frac{1}{2} \cdot 38 \cdot t^2 \]
This simplifies to:
\[ 9.5 = 19 t^2 \]

### Step 4: Solve for \(t^2\)
Rearranging the equation gives:
\[ t^2 = \frac{9.5}{19} \]
Calculating \(t^2\):
\[ t^2 = 0.5 \]

### Step 5: Calculate \(t\)
Taking the square root of both sides to find \(t\):
\[ t = \sqrt{0.5} = \frac{1}{\sqrt{2}} \, \text{seconds} \]

### Final Answer
The time taken by the falling bolt to hit the floor of the elevator is:
\[ t = \frac{1}{\sqrt{2}} \, \text{seconds} \approx 0.707 \, \text{seconds} \]

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