Question

The time taken by a body to slide down the smooth inclined plane is 4sec. The time taken by a body to slide 1/4th of the length of the plane is

• A. 1 sec
• B. 2 sec
• C. 3 sec
• D. 0.5 sec.

Hint:

The relationship \(s \propto t^2\) for motion from rest is very powerful. It implies that \(t \propto \sqrt{s}\). To cover 1/4 of the distance, it will take \(\sqrt{1/4} = 1/2\) of the total time. Half of the total time of 4 seconds is 2 seconds.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
A body starts from rest on a smooth (frictionless) incline. Given the time for the full journey, we need to find the time it takes to cover the first quarter of the distance.

Step 2: Key Formula or Approach:
For an object starting from rest (\(u=0\)) and moving with constant acceleration (\(a\)), the distance covered (\(s\)) in time (\(t\)) is given by the kinematic equation:
\[ s = \frac{1}{2}at^2 \]
From this, we can see that the distance is proportional to the square of the time (\(s \propto t^2\)).

Step 3: Detailed Explanation:
Let \(L\) be the total length of the inclined plane and \(T = 4\) s be the total time to slide down.
Let \(t\) be the time taken to slide a distance of \(s = L/4\).
Using the proportionality \(s \propto t^2\), we can set up a ratio:
\[ \frac{s_1}{s_2} = \frac{t_1^2}{t_2^2} \]
Let \(s_1 = L\), \(t_1 = T = 4\) s.
Let \(s_2 = L/4\), \(t_2 = t\).
\[ \frac{L}{L/4} = \frac{4^2}{t^2} \]
\[ 4 = \frac{16}{t^2} \]
Rearrange to solve for \(t^2\):
\[ t^2 = \frac{16}{4} = 4 \]
\[ t = \sqrt{4} = 2 \text{ s} \]

Step 4: Final Answer:
The time taken to slide 1/4th of the length is 2 seconds.