Q. No.Two particles are separated by a horizontal distance \(x\) as shown in the figure. They are projected at the same time as shown in the figure with different initial speeds. The time after which the horizontal distance between them becomes zero will be:
1.
\(\dfrac{x}{u}\)
2.
\(\dfrac{u}{2 x}\)
3.
\(\dfrac{2 u}{x}\)
4.
None of the above
1. \(4~\text{m/s},\) \(53^\circ\) with \(x\)-axis
2. \(4~\text{m/s},\) \(37^\circ\) with \(x\)-axis
3. \(5~\text{m/s},\) \(53^\circ\) with \(y\)-axis
4. \(5~\text{m/s},\) \(53^\circ\) with \(x\)-axis
A particle starts from the origin at \(t=0\) with a velocity of \(5.0\hat i~\text{m/s}\) and moves in the \(x\text-y\) plane under the action of a force that produces a constant acceleration of \((3.0\hat i + 2.0\hat j)~\text{m/s}^2.\) What is the speed of the particle at the instant its \(x\text-\)coordinate is \(84~\text m?\)
1. \(36~\text{m/s}\)
2. \(26~\text{m/s}\)
3. \(1~\text{m/s}\)
4. Zero
Two boys are standing at the ends \(A\) and \(B\) of the ground where \(AB =a.\) The boy at \(B\) starts running in a direction perpendicular to \(AB\) with velocity \(v_1.\) The boy at \(A\) starts running simultaneously with velocity \(v\) and catches the other boy in a time \(t,\) where \(t\) is:
| 1. | \(\frac{a}{\sqrt{v^2+v^2_1}}\) | 2. | \(\frac{a}{\sqrt{v^2-v^2_1}}\) |
| 3. | \(\frac{a}{v-v_1}\) | 4. | \(\frac{a}{v+v_1}\) |
A stone tied to the end of a \(1\) m long string is whirled in a horizontal circle at a constant speed. If the stone makes \(22\) revolutions in \(44\) seconds, what is the magnitude and direction of acceleration of the stone?
| 1. | \(\pi^2 ~\text{ms}^{-2} \) and direction along the tangent to the circle. |
| 2. | \(\pi^2 ~\text{ms}^{-2} \) and direction along the radius towards the centre. |
| 3. | \(\frac{\pi^2}{4}~\text{ms}^{-2} \) and direction along the radius towards the centre. |
| 4. | \(\pi^2~\text{ms}^{-2} \) and direction along the radius away from the centre. |
A car starts from rest and accelerates at \(5~\text{m/s}^{2}.\) At \(t=4~\text{s}\), a ball is dropped out of a window by a person sitting in the car. What is the velocity and acceleration of the ball at \(t=6~\text{s}?\)
(Take \(g=10~\text{m/s}^2\))
1. \(20\sqrt{2}~\text{m/s}, 0~\text{m/s}^2\)
2. \(20\sqrt{2}~\text{m/s}, 10~\text{m/s}^2\)
3. \(20~\text{m/s}, 5~\text{m/s}^2\)
4. \(20~\text{m/s}, 0~\text{m/s}^2\)
| 1. | \( \theta=\sin ^{-1}\left(\frac{\pi^2 {R}}{{gT}^2}\right)^{1/2}\) | 2. | \(\theta=\sin ^{-1}\left(\frac{2 {gT}^2}{\pi^2 {R}}\right)^{1 / 2}\) |
| 3. | \(\theta=\cos ^{-1}\left(\frac{{gT}^2}{\pi^2 {R}}\right)^{1 / 2}\) | 4. | \(\theta=\cos ^{-1}\left(\frac{\pi^2 {R}}{{gT}^2}\right)^{1 / 2}\) |
1. \(45^{\circ} \)
2. \(90^{\circ} \)
3. \(-45^{\circ} \)
4. \(180^{\circ}\)
If \(\left| \vec{A}\right|\) = \(2\) and \(\left| \vec{B}\right|\) = \(4,\) then match the relations in column-I with the angle \(\theta\) between \(\vec{A}\) and \(\vec{B}\) in column-II.
| Column-I | Column-II |
| (A) \(\left| \vec{A}\times \vec{B}\right|\) \(=0\) | (p) \(\theta=30^\circ\) |
| (B)\(\left| \vec{A}\times \vec{B}\right|\)\(=8\) | (q) \(\theta=45^\circ\) |
| (C) \(\left| \vec{A}\times \vec{B}\right|\) \(=4\) | (r) \(\theta=90^\circ\) |
| (D) \(\left| \vec{A}\times \vec{B}\right|\) \(=4\sqrt2\) | (s) \(\theta=0^\circ\) |
| 1. | A(s), B(r), C(q), D(p) |
| 2. | A(s), B(p), C(r), D(q) |
| 3. | A(s), B(p), C(q), D(r) |
| 4. | A(s), B(r), C(p), D(q) |
| 1. | \(5\sqrt2\) m/s2 SE | 2. | \(\dfrac{5}{\sqrt2}\) m/s2 SE |
| 3. | \(5\sqrt2\) m/s2 NE | 4. | \(\dfrac{5}{\sqrt2}\) m/s2 NE |
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