Nature of motion when distance travelled is directly proportional to t1/2



When the distance travelled by a particle is directly proportional to t1/2, where t is the time elapsed, the nature of motion is said to be under a particular type of motion known as uniformly accelerated motion. In this type of motion, the acceleration is not constant but changes with time.



To better understand the nature of motion, let's break down the equation of distance travelled:

Distance travelled = k * t1/2

where k is a constant of proportionality.

Taking the derivative of distance travelled with respect to time, we get:

Velocity = (d/dt) * (k * t1/2)
Velocity = (1/2) * k * t-1/2

Taking the derivative of velocity with respect to time, we get:

Acceleration = (d/dt) * [(1/2) * k * t-1/2]
Acceleration = (-1/4) * k * t-3/2

From the equation of acceleration, we can see that the acceleration is not constant but changes with time. This is what characterizes uniformly accelerated motion.

The velocity of the particle increases with time, but the rate of increase decreases as time goes on. The same is true for the acceleration of the particle. It increases with time, but the rate of increase decreases as time goes on.



In conclusion, when the distance travelled by a particle is directly proportional to t1/2, where t is the time elapsed, the nature of motion is uniformly accelerated motion. This type of motion is characterized by a changing acceleration that increases with time but at a decreasing rate.

Uniform accelerated motion always see the velocity . after differentiation of x(displacement) wrt to time then see velocity is function of time or not if yes . it must me accelerated motion . no if t has power other than 1 it is a possibility of having non uniform accelerated motion . be careful at that time . so always check the velocity and acceleration .here in your case accelerated motion is uniform because acceleration is not function of time .