The relation between time and distance is $t = \alpha {x^2} + \beta x$, where $\alpha $ and $\beta $ are constants. The retardation is

The acceleration of a moving body can be found from

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 velocity of a bullet is reduced from $200\,m/s$ to $100\,m/s $ while travelling through a wooden block of thickness $10\,cm$. The retardation, assuming it to be uniform, will be 

The displacement of a particle is given by $y = a + bt + c{t^2} - d{t^4}$. The initial velocity and acceleration are respectively

A body $A$ starts from rest with an acceleration $a_1$ . After $2\ seconds$ , another body $B$ starts from rest with an acceleration $a_2$ . If they travel equal distance in the $5th\ second$ , after the start of $A$ , then the ratio $a_1$ : $a_2$ is equal to