A point moves in a straight line under the retardation a${\mathrm{v}}^2$. If the initial velocity is u, the distance covered in 't' seconds is-

1.  $\mathrm{aut}$

2.  $\frac{1}{\mathrm{a}}\ln \left(\mathrm{aut}\right)$

3.  $\frac{1}{\mathrm{a}}\ln \left(1+\mathrm{aut}\right)$

4.  $\mathrm{a} \ln \left(\mathrm{aut}\right)$

A body starts from the origin and moves along the X-axis such that the velocity at any instant is given by $(4t^3-2t)$, where t is in sec and velocity in m/s. What is the acceleration of the particle, when it is 2 m from the origin ?

(1) 28 m/s²

(2) 22 m/s²

(3) 12 m/s²

(4) 10 m/s²

The initial velocity of a particle is u (at t = 0) and the acceleration f is given by at. Which of the following relation is valid 

(1) $v=u+a{t}^2$

(2) $v=u+a\frac{t^2}{2}$

(3) $v=u+at$

(4) v = u

The velocity of a body depends on time according to the equation $v=20+0.1t^2$. The body is undergoing 

(1) Uniform acceleration

(2) Uniform retardation

(3) Non-uniform acceleration

(4) Zero acceleration

A rocket is fired upward from the earth's surface such that it creates an acceleration of 19.6 m/sec². If after 5 sec its engine is switched off, the maximum height of the rocket from earth's surface would be [MP PET 1995]

(1) 245 m

(2) 490 m

(3) 980 m

(4) 735 m

A balloon starts rising from the ground with an acceleration of 1.25 m/s² . After 8s, a stone is released from the balloon. The stone will (g = 10 m/s²) 

(1) Reach the ground in 4 second

(2) Begin to move down after being released

(3) Have a displacement of 50 m

(4) Cover a distance of 40 m in reaching the ground

The acceleration of a particle is increasing linearly with time t as bt. The particle starts from the origin with an initial velocity v₀ The distance travelled by the particle in time t will be 

(1) $v_{0}t+\frac{1}{3}b{t}^2$

(2) $v_{0}t+\frac{1}{3}b{t}^3$

(3) $v_{0}t+\frac{1}{6}b{t}^3$

(4) $v_{0}t+\frac{1}{2}b{t}^2$

The acceleration of a particle starting from rest, varies with time according to the relation A = – aω² sinω t. The displacement of this particle at a time t will be

(1) $-\frac{1}{2}\left(a{\omega }^2\sin \omega t\right)t^2$

(2) $a\omega \sin \omega t$

(3) $a\omega \cos \omega t$

(4) $a\sin \omega t$