In a straight line motion the distance travelled is proportional to the square root of Time taken. The acceleration of a particle is proportional to:-(i)√v(ii) v^2(iii)velocity (iv)v^3.Why?

Cube of velocity

Cube of the velocity

To determine which relationship is proportional to the acceleration of a particle, we can use the information given about the distance traveled.

According to the given information, in a straight line motion, the distance traveled is proportional to the square root of the time taken. Mathematically, we can express this relationship as:

Distance ∝ √Time

Differentiating this equation with respect to time (t), we can find the expression for velocity (v):

Velocity ∝ d(Distance)/dt = d(√Time)/dt

Using the chain rule of differentiation, we get:

Velocity ∝ 1 / (2√Time) * d(Time)/dt

Simplifying further:

Velocity ∝ 1 / (2√Time)

Therefore, we can conclude that the velocity of a particle is inversely proportional to the square root of the time taken.

Now, to find the relationship between acceleration and velocity, we need to differentiate the velocity equation we derived earlier. Differentiating the previous equation with respect to time (t), we get:

Acceleration ∝ d(Velocity)/dt = d(1 / (2√Time))/dt

Using the chain rule of differentiation, we get:

Acceleration ∝ -1 / (4√Time^3) * d(Time)/dt

Simplifying further:

Acceleration ∝ -1 / (4√Time^3)

Thus, we can conclude that the acceleration of a particle is proportional to 1 / (4√Time^3).