Displacement x of a particle moving in one dimensional motion under the action of a constant force is given as t = x^1/2 + 3 wher x is in m and t is in s . Dispacement of particle wen velocity is zero canb?

To find the displacement of the particle when its velocity is zero, we first need to analyze the relationship between displacement and time given by the equation \( t = x^{1/2} + 3 \). This equation shows how displacement \( x \) varies with time \( t \). Let's break this down step by step.

Understanding the Relationship

The equation can be rearranged to express displacement \( x \) in terms of time \( t \). Starting from:

t = x^{1/2} + 3

We can isolate \( x^{1/2} \):

x^{1/2} = t - 3

Next, squaring both sides gives us:

x = (t - 3)^2

Finding Velocity

Velocity \( v \) is defined as the rate of change of displacement with respect to time. Mathematically, this is expressed as:

v = dx/dt

To find \( v \), we need to differentiate \( x \) with respect to \( t \). Using the chain rule, we have:

v = d/dt[(t - 3)^2]

Applying the power rule:

v = 2(t - 3)(1) = 2(t - 3)

Setting Velocity to Zero

To find the time when the velocity is zero, we set \( v = 0 \):

0 = 2(t - 3)

This simplifies to:

t - 3 = 0

Thus, we find:

t = 3 \text{ seconds}

Calculating Displacement at Zero Velocity

Now that we have the time when the velocity is zero, we can substitute \( t = 3 \) back into our equation for displacement:

x = (3 - 3)^2

This simplifies to:

x = 0^2 = 0 \text{ meters}

Final Thoughts

Therefore, the displacement of the particle when its velocity is zero is 0 meters. This indicates that at the moment when the particle stops moving, it is at the origin of our coordinate system. This scenario is a good example of how displacement, velocity, and time are interrelated in one-dimensional motion under constant force.