The relation between time and distance is given by $$\(t=\alpha x^2+\beta x,\) where \(\alpha\) and \(\beta\) are constants. The retardation, as calculated based on this equation, will be (assume \(v\) to be velocity):
1. $\mathrm{}$\(2\alpha v^3\)
2. \(2\beta v^3\)$\mathrm{}{\mathrm{}}^{}$
3. \(2\alpha\beta v^3\)$\mathrm{}{\mathrm{}}^{}$
4. \(2\beta^2 v^3\)$\mathrm{}{\mathrm{}}^{}$

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

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 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

The motion of a particle is described by the equation $x=a+b{t}^2$ where a = 15 cm and b = 3 cm/s². Its instantaneous velocity at time 3 sec will be 

1. 36 cm/sec

2. 18 cm/sec

3. 16 cm/sec

4. 32 cm/sec