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

1. $2\alpha v^3$

2. $2\beta v^3$

3. $2\alpha \beta v^3$

4. $2{\beta }^2{v}^3$

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

Velocity of a particle changes when 

(1) Direction of velocity changes

(2) Magnitude of velocity changes

(3) Both of above

(4) None of the above

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: 

Consider the acceleration, velocity and displacement of a tennis ball as it falls to the ground and bounces back. Directions of which of these changes in the process ?

1. Velocity only

2. Displacement and velocity

3. Acceleration, velocity and displacement

4. Displacement and acceleration

A particle moves along X-axis in such a way that its coordinate X varies with time t according to the equation $x=(2-5t+6t^2)m$. The initial velocity of the particle is 

(1) –5 m/s

(2) 6 m/s

(3) –3 m/s

(4) 3 m/s 

Starting from rest, acceleration of a particle is $a=2(t-1).$ The velocity of the particle at $t=5s$ is:

1.  15 m/sec
2.  25 m/sec
3.  5 m/sec
4.  None of these