Q. No.A particle starting from the point \((1,2)\) moves in a straight line in the \(XY\)-plane. Its coordinates at a later time are \((2,3).\) The path of the particle makes what angle with the \(x\)-axis?
1. \(30^\circ\)
2. \(45^\circ\)
3. \(60^\circ\)
4. data is insufficient
1. \(30^\circ\)
2. \(45^\circ\)
3. \(60^\circ\)
4. data is insufficient
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A particle is projected from a horizontal plane (\(x\text-z\) plane) such that its velocity vector at time \(t\) is given by \(\vec{v}=a \hat{i}+(b-c t )\hat{j}\). Its range on this horizontal plane is given by:
| 1. | \(\frac{ba}{c} \) | 2. | \(\frac{2ba}{c} \) |
| 3. | \(\frac{3ba}{c} \) | 4. | None |
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1. \(45^\circ\)
2. \(30^\circ\)
3. \(60^\circ\)
4. \(90^\circ\)
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A river is \(1\) km wide. The banks are straight and parallel. The current is \(5\) km/h and is parallel to the banks. A boat has a maximum speed of \(3\) km/h in still water. In what direction should the boat head so as to arrive at point \(B\) directly opposite to its starting point \(A\)?
| 1. | directly across the river. |
| 2. | head \(53^{\circ}\) upstream from the line \(AB\). |
| 3. | head \(37^{\circ}\) upstream from the line \(AB\). |
| 4. | the trip from \(A\) to \(B\) is not possible with this speed. |
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Two particles \(A\) and \(B,\) move with constant velocities \(\vec{v_1}\) and \(\vec{v_2}.\) At the initial moment their position vector are \(\vec {r_1}\) and \(\vec {r_2}\) respectively. The conditions for particles \(A\) and \(B\) for their collision to happen will be:
| 1. | \(\vec{r_{1 }} . \vec{v_{1}} = \vec{r_{2 }} . \vec{v_{2}}\) | 2. | \(\vec{r_{1}} \times\vec{v_{1}} = \vec{r_{2}} \times \vec {v_{2}}\) |
| 3. | \(\vec{r_{1}}-\vec{r_{2}}=\vec{v_{1}} - \vec{v_{2}}\) | 4. | \(\frac{\vec{r_{1}} - \vec{r_{2}}}{\left|\vec{r_{1}} - \vec{r_{2}}\right|} = \frac{\vec{v_{2}} - \vec{v_{1}}}{\left|\vec{v_{2}} - \vec{v_{1}}\right|}\) |
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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 \(\frac{1}{3}~\text{m/s}\). 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. \(2\hat j~\text{m/s}^2\)
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Three balls are thrown from the top of a building with equal speeds at different angles. When the balls strike the ground, their speeds are \(v_{1} , v_{2}\) \(\text{and}\) \(v_{3}\) respectively, then:
1. \(v_{1} > v_{2} > v_{3}\)
2. \(v_{3} > v_{2} = v_{1}\)
3. \(v_{1} = v_{2} = v_{3}\)
4. \(v_{1} < v_{2} < v_{3}\)
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Figure below shows a body of mass \(M\) moving with a uniform speed \(v\) on a circular path of radius, \(R\). What is the change in acceleration in going from \(P_1\) to \(P_2\)?
1. zero
2. \(v^{2} / 2 R\)
3. \(2 v^{2} / R\)
4. \(\frac{v^{2}}{R} \times \sqrt{2}\)
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A particle is projected with a speed \(u\) at an angle \(\theta\) to the horizontal. Radius of curvature at highest point of its trajectory is?
1. \(\frac{u^{2} \cos^{2} \theta}{2 g}\)
2. \(\frac{\sqrt{3} u^{2}\cos^{2} \theta}{2 g}\)
3. \(\frac{u^{2} \cos^{2} \theta}{g}\)
4. \(\frac{\sqrt{3} u^{2} \cos^{2} \theta}{g}\)
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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:
| 1. | \( 1: 1 \) | 2. | \(r_A: r_B \) |
| 3. | \(v_A: v_B \) | 4. | \(r_B: r_A\) |
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