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. | \(- 2 nβ^{2} x^{- 4 n + 1}\) |
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. \(\frac{1}{2}f_0T\)
The displacement of a particle is given by \(y = a + bt + ct^{2} - dt^{4}\). The initial velocity and acceleration are, respectively:
1. | \(b, -4d\) | 2. | \(-b,2c\) |
3. | \(b, ~2c\) | 4. | \(2c, -2d\) |
The acceleration \(a\) (in ) 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 point moves in a straight line under the retardation \(av^2\). 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. \(a~\mathrm{ln}(aut)\)
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. |