A particle moving with a uniform acceleration travels \(24\) m and \(64\) m in the first two consecutive intervals of \(4~\text{s}\) each. Its initial velocity is:
1. \(1~\text{m/s}\)
2. \(10~\text{m/s}\)
3. \(5~\text{m/s}\)
4. \(2~\text{m/s}\)

The position of a particle moving in the XY plane at any time t is given by $x=(3t^2-6t)$ metres. Select the correct statement about the moving particle from the following.

1. The acceleration of the particle is zero at t = 0 second
2. The velocity of the particle is zero at t = 0 second
3. The velocity of the particle is zero at t = 1 second
4. The velocity and acceleration of the particle are never zero

Two cars \(A\) and \(B\) are travelling in the same direction with velocities \(v_1\) and \(v_2 (v_1>v_2)\). When the car \(A\) is at a distance \(d\) behind car \(B\), the driver of the car \(A\) applied the brake producing uniform retardation \(a\). There will be no collision when:
1. \(d< \dfrac{(v_1-v_2)^2}{2a}\)

2. \(d< \dfrac{v^2_1-v^2_2}{2a}\)

3. \(d> \dfrac{(v_1-v_2)^2}{2a}\)

4. \(d> \dfrac{v^2_1-v^2_2}{2a}\)

A body moves from rest with a constant acceleration of 5 m/s². Its instantaneous speed (in m/s) at the end of 10 sec is  

1. 50

2. 5

3. 2

4. 0.5

The acceleration \(a\) in m/s² of a particle is given by $a=3t^2+2t+2$ where t is the time. If the particle starts out with a velocity, \(u=2\) m/s at t = 0, then the velocity at the end of \(2\) seconds will be:
1. \(12\) m/s
2. \(18\) m/s
3. \(27\) m/s
4. \(36\) m/s

The displacement of a particle starting from rest (at t = 0) is given by $s=6t^2-t^3$. The time in seconds at which the particle will attain zero velocity again, is  

1. 2

2. 4

3. 6

4. 8

A body is moving according to the equation $x=at+b{t}^2-c{t}^3$ where x = displacement and a, b and c are constants. The acceleration of the body is 

1. $a+2bt$ 

2. $2b+6ct$ 

3. $2b-6ct$ 

4. $3b-6c{t}^2$ 

The relation \(3t = \sqrt{3x} + 6\) describes the displacement of a particle in one direction where \(x\) is in metres and \(t\) in seconds. The displacement, when velocity is zero, is: 

The average velocity of a body moving with uniform acceleration travelling a distance of 3.06 m is 0.34 ms^–1. If the change in velocity of the body is 0.18ms^–1 during this time, its uniform acceleration is 

1. 0.01 ms^–2

2. 0.02 ms^–2

3. 0.03 ms^–2

4. 0.04 ms^–2

The displacement of a particle is proportional to the cube of time elapsed. How does the acceleration of the particle depends on time obtained 

1. $a\propto t^2$

2. $a\propto t^4$

3. $a\propto t^3$

4. $a\propto t$