Question

give
type of motion of object if( i)its displacement is proportional to the time interval and( ii) displacement is proportional to the square of time​

Answer 1

First condition says the displacement is directly proportional to time, i.e.,

$\sf{\longrightarrow s\propto t}$

Let,

$\sf{\longrightarrow s=kt}$

where $\sf{k}$ is a time independent constant.

Differentiating wrt $\sf{t,}$ we get the velocity.

$\sf{\longrightarrow v=k}$

It is clear that the velocity is independent of time, so the object moves with constant velocity throughout the time.

So it undergoes a uniform motion here.

Second condition says the displacement is directly proportional to square of time, i.e.,

$\sf{\longrightarrow s\propto t^2}$

Let,

$\sf{\longrightarrow s=kt^2}$

where $\sf{k}$ is a time independent constant.

Differentiating wrt $\sf{t,}$ we get the velocity.

$\sf{\longrightarrow v=2kt}$

Again differentiating wrt $\sf{t,}$ we get the acceleration.

$\sf{\longrightarrow a=2k}$

It is clear that the acceleration is independent of time, so the object moves with constant acceleration or uniformly varying velocity throughout the time.

So it undergoes a uniformly accelerated motion here.