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 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 particle moving along the x-axis has acceleration \(f,\) at time \(t,\) given by, \(f=f_0\left ( 1-\frac{t}{T} \right ),\)  where \(f_0\) and \(T\) are constants. The particle at \(t=0\) has zero velocity. In the time interval between \(t=0\) and the instant when \(f=0,\) the particle’s velocity \( \left ( v_x \right )\) is:
1. \(f_0T\)
2. \(\frac{1}{2}f_0T^{2}\)
3. \(f_0T^2\)
4. \(\frac{1}{2}f_0T\)