A very broad elevator is going up vertically with a constant acceleration of `2m//s^(2)`. At the instant when its velocity is `4m//s` a ball is projected form the floor of the lift with a speed of `4m//s` relative to the floor at an elevation of `30^(@)`. Time taken by the ball to return the floor is `(g=10ms^(2))`

Options

A. `1/2s`
B. `1/3s`
C. `1/4s`
D. `1s`

Explanation

To solve the problem, we need to analyze the motion of the ball projected from the elevator. Here's a step-by-step solution:

### Step 1: Understand the given data
- The elevator is accelerating upwards with an acceleration \( a_{lift} = 2 \, \text{m/s}^2 \).
- The velocity of the elevator at the time of projection is \( v_{lift} = 4 \, \text{m/s} \).
- The ball is projected with a speed \( v_{ball} = 4 \, \text{m/s} \) at an angle of \( \theta = 30^\circ \) relative to the floor of the elevator.
- The acceleration due to gravity is \( g = 10 \, \text{m/s}^2 \).

### Step 2: Resolve the velocity of the ball into components
The velocity of the ball can be resolved into horizontal and vertical components:
- **Horizontal component** \( v_{x} = v_{ball} \cdot \cos(\theta) = 4 \cdot \cos(30^\circ) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} \, \text{m/s} \)
- **Vertical component** \( v_{y} = v_{ball} \cdot \sin(\theta) = 4 \cdot \sin(30^\circ) = 4 \cdot \frac{1}{2} = 2 \, \text{m/s} \)

### Step 3: Determine the effective acceleration
Since the elevator is accelerating upwards, the effective acceleration acting on the ball will be the sum of the gravitational acceleration and the elevator's acceleration:
\[ a_{effective} = g + a_{lift} = 10 \, \text{m/s}^2 + 2 \, \text{m/s}^2 = 12 \, \text{m/s}^2 \]

### Step 4: Calculate the time of flight
The time of flight \( t \) for a projectile launched vertically can be calculated using the formula:
\[ t = \frac{2 v_{y}}{a_{effective}} \]
Substituting the values:
\[ t = \frac{2 \cdot 2}{12} = \frac{4}{12} = \frac{1}{3} \, \text{s} \]

### Conclusion
The time taken by the ball to return to the floor of the elevator is \( \frac{1}{3} \, \text{s} \) or approximately \( 0.33 \, \text{s} \).

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