A body slides down a smooth plane starting from rest in 4sec. Time taken to slide first 1/4 of the distance is

Options

A. 2 s
B. 1 s
C. 3 s
D. (1/8) s

Answer

To solve the problem, we need to find the time taken for a body to slide down the first quarter of the distance on a smooth inclined plane, given that it takes a total of 4 seconds to slide down the entire distance from rest.

### Step-by-Step Solution:

1. **Understanding the Motion**:
The body starts from rest and slides down a smooth incline. The total time taken to slide down the entire distance is given as 4 seconds.

2. **Using the Equation of Motion**:
We can use the second equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
where:
- \( s \) is the total distance,
- \( u \) is the initial velocity (which is 0 since it starts from rest),
- \( a \) is the acceleration down the incline,
- \( t \) is the total time (4 seconds).

Since the body starts from rest, the equation simplifies to:
\[ s = \frac{1}{2} a t^2 \]
Substituting \( t = 4 \) seconds:
\[ s = \frac{1}{2} a (4^2) = 8a \]

3. **Finding the Distance for the First Quarter**:
The distance for the first quarter of the total distance is:
\[ s_1 = \frac{1}{4} s = \frac{1}{4} (8a) = 2a \]

4. **Using the Same Equation for the First Quarter**:
Now, we need to find the time taken to slide this distance \( s_1 = 2a \). Using the same equation of motion:
\[ s_1 = ut + \frac{1}{2} a t_1^2 \]
Here, \( u = 0 \), so:
\[ 2a = \frac{1}{2} a t_1^2 \]
Simplifying this gives:
\[ 2a = \frac{1}{2} a t_1^2 \implies 4 = t_1^2 \implies t_1 = 2 \text{ seconds} \]

5. **Conclusion**:
The time taken to slide down the first quarter of the distance is **2 seconds**.

### Final Answer:
The time taken to slide the first quarter of the distance is **2 seconds**.