At any instant, the velocity and acceleration of a particle moving along a straight line are v and a. The speed of the particle is increasing if

(1) v>0, a>0

(2) v<0, a>0

(3) v>0, a<0

(4) v>0, a=0

If v is the velocity of a body moving along x-axis, then acceleration of body is

(1) $\frac{dv}{dx}$

(2) $v\frac{dv}{dx}$

(3) $x\frac{du}{dx}$

(4) $v\frac{dx}{dv}$

If a body is moving with constant speed, then its acceleration

(1) Must be zero

(2) May be variable

(3) May be uniform

(4) Both (2) & (3)

An object is moving with variable speed, then

(1) Its velocity may be zero

(2) Its velocity must be variable

(3) Its acceleration may be zero

(4) Its velocity must be constant

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 $x=A \sin \left(\omega t\right)$ 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 x² = 2 + t. Its acceleration is given by

(1) $\frac{-2}{x^3}$

(2) $-\frac{1}{2x^2}$

(3) $-\frac{1}{4x^3}$

(4) $\frac{1}{x^2}$

A body is moving with variable acceleration (a) along a straight line. The average acceleration of the body in time interval t₁ to t₂ 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) $\frac{{\int }_{t_1}^{t_2}adt}{t_2-t_1}$

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. $\frac{v_1+v_2}{t_1-t_2}$