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

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 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:

A body is projected vertically upward with a speed 10 m/s and another at the same time with the same speed in the downward direction from the top of a tower. The magnitude of acceleration of first body with respect to second is {take g = 10 m/s²}

(1) Zero

(2) 10 m/s²

(3) 5 m/s²

(4) 20 m/s²

A car with a vertical windshield moves in a rain storm at a speed of \(40\) km/hr. The rain drops fall vertically with a constant speed of \(20\) m/s. The angle at which raindrops strike the windshield is:
1. \(\tan^{- 1} \frac{5}{9}\)
2. \(\tan^{- 1} \frac{9}{5}\)
3. \(\tan^{- 1} \frac{3}{2}\)
4. \(\tan^{- 1} \frac{2}{3}\)

Three particles A, B, and C are projected from the same point with same initial speeds making angles 30^o, 45^o, and 60^o, respectively, with the horizontal. Which of the following statements is correct?

1. A and C do not have equal ranges.

2. Ranges of A and C are equal and less than that of B.

3. Ranges of A and C are equal and greater than that of B.

4. A, B and C have equal ranges.

The horizontal range of a projectile fired at an angle of 15^o is 50 m. If it is fired with the same speed at an angle of 45^o, its range will be:

1. 60 m

2. 71 m

3. 100 m

4. 141 m