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

A particle is moving such that its position coordinates (x, y) are (\(2\) m, \(3\) m) at time \(t=0,\) (\(6\) m,\(7\) m) at time \(t=2\) s, and (\(13\) m, \(14\) m) at time \(t=\) \(5\) s. The average velocity vector \(\vec{v}_{avg}\) from \(t=\) 0 to \(t=\) \(5\) s is:
1. \({1 \over 5} (13 \hat{i} + 14 \hat{j})\)
2. \({7 \over 3} (\hat{i} + \hat{j})\)
3. \(2 (\hat{i} + \hat{j})\)
4. \({11 \over 5} (\hat{i} + \hat{j})\)

A stone falls freely under gravity. It covers distances \(h_1,~h_2\) and \(h_3\) in the first \(5\) seconds, the next \(5\) seconds and the next \(5\) seconds respectively. The relation between \(h_1,~h_2\) and \(h_3\) is:

A particle moves a distance \(x\) in time \(t\) according to equation \(x=(t+5)^{-1}.\) The acceleration of the particle is proportional to:
1. (velocity)^\(3/2\)
2. (distance)^\(2\)
3. (distance)^\(-2\)
4. (velocity)^\(2/3\)

A particle starts its motion from rest under the action of a constant force. If the distance covered in the first \(10\) s is \(S_1\) and that covered in the first \(20\) s is \(S_2\), then:
1. \(S_2=2S_1\)
2. \(S_2 = 3S_1\)
3. \(S_2 = 4S_1\)
4. \(S_2= S_1\)

A particle shows the distance-time curve as given in this figure. The maximum instantaneous velocity of the particle is around the point:

         

The position of a particle with respect to time \(t\) along the \(\mathrm{x}\)-axis is given by \(9t^{2}-t^{3}\) where x is in metre and \(t\) in second. What will be the position of this particle when it achieves maximum speed along the \(+\mathrm{x}\) direction?
1. \(32\) m
2. \(54\) m
3. \(81\) m
4. \(24\) m

A car moves from \(\mathrm{X}\) to \(\mathrm{Y}\) with a uniform speed \(\mathrm{v_u}\) and returns to \(\mathrm{X}\) with a uniform speed \(\mathrm{v_d}.\) The average speed for this round trip is:

1. $\frac{2{\mathrm{v}}_{\mathrm{d}}{\mathrm{v}}_{\mathrm{u}}}{{\mathrm{v}}_{\mathrm{d}}+{\mathrm{v}}_{\mathrm{u}}}$

2. $\sqrt{{\mathrm{v}}_{\mathrm{u}}{\mathrm{v}}_{\mathrm{d}}}$

3. $\frac{{\mathrm{v}}_{\mathrm{d}}{\mathrm{v}}_{\mathrm{u}}}{{\mathrm{v}}_{\mathrm{d}}+{\mathrm{v}}_{\mathrm{u}}}$

4. $\frac{{\mathrm{v}}_{\mathrm{u}}+{\mathrm{v}}_{\mathrm{d}}}{2}$