The relation between time and distance is $\mathrm{t}={\mathrm{\alpha x}}^2+\mathrm{\beta x}$, where $\mathrm{\alpha }$ and $\mathrm{\beta }$ are constants. The retardation is

1.  $2{\mathrm{\alpha v}}^3$

2.  $2{\mathrm{\beta v}}^3$

3.  $2{\mathrm{\alpha \beta v}}^3$

4.  $2{\mathrm{\beta }}^2{\mathrm{v}}^3$

A point moves in a straight line under the retardation ${\mathrm{av}}^2$. If the initial velocity is u, the distance covered in 't' second is-

1.  aut

2.  $\frac{1}{\mathrm{a}}\ln \left(\mathrm{aut}\right)$

3.  $\frac{1}{\mathrm{a}}\ln \left(1+\mathrm{aut}\right)$

4.  $\mathrm{a} \ln \left(\mathrm{aut}\right)$

A particle moves along a parabolic path \(y =9x^2\) in such a way that the \(x\) component of the velocity remains constant and has a value of \(\frac{1}{3}~\text{m/s}\)${\mathrm{}}^{}$. It can be deduced that the acceleration of the particle will be:
1. \(\frac{1}{3}\hat j~\text{m/s}^2\)
2. \(3\hat j~\text{m/s}^2\)
3. \(\frac{2}{3}\hat j~\text{m/s}^2\)
4. \(2\hat j~\text{m/s}^2\)

The given graph shows the variation of velocity with displacement. Which one of the graph given below correctly represents the variation of acceleration with displacement?

1.  

2.  

3.  

4.  

A particle starts from rest. Its acceleration (a) versus time (t) is as shown in the figure. The maximum speed of the particle will be

1.  $110 {\mathrm{ms}}^{-1}$

2.  $55 {\mathrm{ms}}^{-1}$

3.  $550 {\mathrm{ms}}^{-1}$

4.  $660 {\mathrm{ms}}^{-1}$

A particle starts from rest at t=0 and undergoes an acceleration a in ms^-2 with time t in second which is as shown

Which one of the following plot represents velocity v in ms^-1 versus time t in second?

1.  

2.  

3.  

4.  

Two particles \(\mathrm{A}\) and \(\mathrm{B}\), move with constant velocities \(\overrightarrow{{v}_1}\) and \(\overrightarrow{{v}_2}\) respectively. At the initial moment, their position vectors are \(\overrightarrow{{r}_1}\) and \(\overrightarrow{{r}_2}\) respectively. The condition for particles \(\mathrm{A}\) and \(\mathrm{B}\) for their collision will be:
1. \(\vec{r_1} \cdot \vec{v_1}=\vec{r_2} \cdot \vec{v_2}\)
2. ​​​​​\(\dfrac{\vec{r_1}-\vec{r_2}}{\left|\vec{r_1}-\vec{r_2}\right|}=\dfrac{\vec{v_2}-\vec{v_1}}{\left|\vec{v_2}-\vec{v_1}\right|}\)${\mathrm{}}_{}$
3. \(\vec{r_1} \times \vec{v_1}=\vec{r_2} \times \vec{v_2}\)${\mathrm{ \; \; }}_{}$
4. \(\vec{r_1}-\vec{r_2}=\vec{v_1}-\vec{v_2}\) 

Two particles \(A\) and \(B\) are moving in a uniform circular motion in concentric circles of radii \(r_A\) and \(r_B\) with speeds \(v_A\) and \(v_B\) respectively. Their time periods of rotation are the same. The ratio of the angular speed of \(A\) to that of \(B\) will be:

The speed of a swimmer in still water is \(20~\text{m/s}.\) The speed of river water is \(10~\text{m/s}\) and is flowing due east. If he is standing on the south bank and wishes to cross the river along the shortest path, the angle at which he should make his strokes with respect to the north is given by:

A point moves in a straight line so that its displacement x metre at time t sec is given by ${\mathrm{x}}^2=1+{\mathrm{t}}^2$ . Its acceleration in m/s² at time t sec is

1. $\frac{1}{{\mathrm{x}}^3}$

2. $\frac{1}{\mathrm{x}}-\frac{1}{x^2}$

3. $\frac{1}{\mathrm{x}}-\frac{t^2}{x^3}$

4. $\frac{-\mathrm{t}}{{\mathrm{x}}^2}$