Question

The displacement \( x \) of a particle moving in one direction is given by \( t = \sqrt{x} + 3 \), where \( x \) is in meters and \( t \) is in seconds. Its displacement when its velocity becomes zero is:

• A. 3 m
• B. 2 m
• C. 1 m
• D. Zero

Hint:

To find when the velocity is zero, differentiate the displacement equation with respect to time and set the derivative equal to zero. This will give you the time when the velocity is zero. Then substitute this time back into the displacement equation to find the displacement at that time.

Answer 1

The Correct Option is D

Given the equation for displacement in terms of time:

\( t = \sqrt{x} + 3 \)

where \( t \) is the time and \( x \) is the displacement.

To find the displacement when the velocity is zero, we need to find the velocity first, which is the rate of change of displacement with respect to time.

The velocity \( v \) is given by the derivative of \( x \) with respect to \( t \):

\( v = \frac{dx}{dt} \)

First, rearrange the given equation to express \( x \) as a function of \( t \):

\( t - 3 = \sqrt{x} \quad \Rightarrow \quad x = (t - 3)^2 \)

Now differentiate \( x = (t - 3)^2 \) with respect to time \( t \):

\( \frac{dx}{dt} = 2(t - 3) \)

To find when the velocity becomes zero, set \( \frac{dx}{dt} = 0 \):

\( 2(t - 3) = 0 \quad \Rightarrow \quad t = 3 \)

Now substitute \( t = 3 \) into the displacement equation \( x = (t - 3)^2 \):

\( x = (3 - 3)^2 = 0 \)

Conclusion: The displacement when the velocity becomes zero is zero. Thus, the correct answer is (4) Zero.