The position of a particle moving along x-axis is given by $x=10t-2t^2$. Then the time (t) at which it will momently come to rest is

1. 0

2. 2.5 s

3. 5 s

4. 10 s

If the displacement of a particle varies with time as $\sqrt{x}=t+7$, then

(1) Velocity of the particle is inversely proportional to t

(2) Velocity of the particle is proportional to t²

(3) Velocity of the particle is proportional to $\sqrt{t}$

(4) The particle moves with constant acceleration.

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) $\sqrt{10}u$