The displacement of a particle is given by \(y = a + bt + ct^{2} - dt^{4}\). The initial velocity and acceleration are, respectively:

The motion of a particle along a straight line is described by the equation
\(x=8+12 t-t^3\) where \(x\) is in metre and \(t\) is in second. The retardation of the particle when its velocity becomes zero is:

The acceleration \(a\) (in ${\mathrm{ms}}^{-2}$) of a body, starting from rest varies with time \(t\) (in \(\mathrm{s}\)) as per the equation \(a=3t+4.\) The velocity of the body at time \(t=2\) \(\mathrm{s}\) will be:
1. \(10~\text{ms}^{-1}\)
2. \(18~\text{ms}^{-1}\)
3. \(14~\text{ms}^{-1}\)
4. \(26~\text{ms}^{-1}\)

A particle of unit mass undergoes one-dimensional motion such that its velocity varies according to \(v(x)= βx^{- 2 n}\) where \(\beta\) and \(n\) are constants and \(x\) is the position of the particle. The acceleration of the particle as a function of \(x\) is given by:
1. \(- 2 nβ^{2} x^{- 2 n - 1}\)
2. \(- 2 nβ^{2} x^{- 4 n - 1}\)
3. \(- 2 \beta^{2} x^{- 2 n + 1}\)
4. \(- 2 nβ^{2} x^{- 4 n + 1}\)
${\mathrm{}}^{}$

The acceleration of a particle starting from rest varies with time according to the relation \(A= - a\omega^2\sin \omega t\). The displacement of this particle at a time \(t\) will be:
1. \(-\frac{1}{2}\left(a\omega^2\sin\omega t\right)t^2\)
2. \(a\omega \sin \omega t\)
3. \(a\omega \cos \omega t\)
4. \(a\sin \omega t\)​​​​​​​

For the given acceleration \(\left ( a \right )\) versus time \(\left ( t \right )\) graph of a body, the body is initially at rest.

       
From the following, the velocity \(\left ( v \right )\) versus time \(\left ( t \right )\) graph will be:

1.		2.
3.		4.

A particle is moving along the \(x\)-axis such that its velocity varies with time as per the equation \(v = 20\left(1-\frac{t}{2}\right)\). At \(t=0\) particle is at the origin. From the following, select the correct position \((x)\) - time \((t)\) plot for the particle:

1.		2.
3.		4.

A point moves in a straight line under the retardation \(av^2\)${\mathrm{}}^{}$. If the initial velocity is \(u,\) the distance covered in \(t\) seconds is:
1. \((aut)\)
2. \(\frac{1}{a}\mathrm{ln}(aut)\)
3. \(\frac{1}{a}\mathrm{ln}(1+aut)\)
4. \(a~\mathrm{ln}(aut)\)

A body is thrown vertically upwards. If the air resistance is to be taken into account, then the time during which the body rises is: 

The initial velocity of a particle is \(u\) (at \(t=0\)) and the acceleration \(f\) is given by \(at\). Which of the following relation is valid?
1. \(v = u + a t^{2}\)
2. \(v = u + a \frac{t^{2}}{2}\)
3. \(v = u + a t\)
4. \(v= u\)