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

(1) $v=u+\alpha t^{3/2}$

(2) $v=u+\frac{3\alpha t^3}{2}$

(3) $v=u+\frac{2}{5}\alpha t^{5/2}$

(4) $v=u+\alpha t^{5/2}$

The position x of a particle moving along the x-axis varies with time t as $x=A<mspace\; width="1em"></mspace>\sin \left(\omega t\right)$ where A and $\omega$ are positive constants. The acceleration a of the particle varies with its position (x) as

1. a = Ax

2. $a=-{\omega }^{2}x$

3. $a=A\omega x$

4. $a={\omega }^2\times A$

A particle moves in a straight line and its position \(x\) at time \(t \)is given by \(x^2 = 2 + t.\) Its acceleration is given by
1. \(-\dfrac{2}{x^3}\)
2. \(-\dfrac{1}{2x^2}\)
3. \(-\dfrac{1}{4x^3}\)
4. \(\dfrac{1}{x^2}\)

A body is moving with variable acceleration \((a)\) along a straight line. The average acceleration of the body in time interval \(t_1\) to \(t_2\) is

(1) $\frac{a\left[t_2+t_1\right]}{2}$

(2) $\frac{a\left[t_2-t_1\right]}{2}$

(3) $\frac{{\int }_{t_1}^{t_2}adt}{t_2+t_1}$

(4) $\frac{{\int }_{t_1}^{t_2}adt}{t_2-t_1}$

The position of a particle moving along the x-axis given by $x=\left(-2t^3+3t^2+5\right)m$. The acceleration of a particle at the instant its velocity becomes zero is

(1) 12 m/s²

(2) -12 m/s²

(3) -6  m/s²

(4) Zero

A particle moves with velocity \(v_1\) for time \(t_1\) and \(v_2\) for time \(t_2\) along a straight line. The magnitude of its average acceleration is:
1. $\frac{v_2-v_1}{t_1-t_2}$
2. $\frac{v_2-v_1}{t_1+t_2}$
3. $\frac{v_2-v_1}{t_2-t_1}$
4. $\frac{v_1+v_2}{t_1-t_2}$

A particle starts moving with acceleration 2 m/s². Distance travelled by it in 5^th half-second is-

(1) 1.25 m

(2) 2.25 m

(3) 6.25 m

(4) 30.25 m

The two ends of a train moving with constant acceleration pass a certain point with velocities u and 3u. The velocity with which the middle point of the train passes the same point is

(1) 2u

(2) $\frac{3}{2}u$

(3) $\sqrt{5}u$

(4) $\sqrt{10}u$

A train starts from rest from a station with acceleration \(0.2 \text{ m/s}^2\) on a straight track and then comes to rest after attaining maximum speed on another station due to retardation \(0.4 \text{ m/s}^2.\) If total time spent is half an hour, then the distance between two stations is- [Neglect length of train]

1. \(216\text{ km}\)

2. \(512\text{ km}\)

3. \(728\text{ km}\)

4. \(1296\text{ km}\)

A body is projected vertically in the upward direction from the surface of the earth. If the upward direction is taken as positive, then the acceleration of the body during its upward and downward journey is: