The velocity-time and acceleration-time graphs of a particle are given as Its position-time graph may be gvien as

Draw $x \to t$ graph for zero acceleration.

The initial velocity of a particle moving along $x$-axis is $u$ (at $t=0$ and $x=0$ ) and its acceleration $a$ is given by $a=k x$. Which of the following equation is correct between its velocity $(v)$ and position $(x)$ ?

Given below are two statements:

Statement $I:$ Area under velocity- time graph gives the distance travelled by the body in a given time.

Statement $II:$ Area under acceleration- time graph is equal to the change in velocity- in the given time.

In the light of given statements, choose the correct answer from the options given below.

The initial velocity of a particle is $u\left(\right.$ at $t=0$ ) and the acceleration a is given by $\alpha t^{3 / 2}$. Which of the following relations is valid?

A particle is moving in a straight line with initial velocity and uniform acceleration $a$. If the sum of the distance travelled in $t^{\text {th }}$ and $( t +1)^{ th }$ seconds is $100 cm$, then its velocity after $t$ seconds, in $.........cm / s$, is