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Q. No.A particle is released from the top of a smooth hemisphere of radius \(R,\) and it slides down along its surface. After it slides down a height \(\frac R5,\) its acceleration will be \(a,\) where:
1.
\(a<\dfrac{2 g}{5}\)
2.
\(\dfrac{2 g}{5}< a< \dfrac{3 g}{5}\)
3.
\(\dfrac {3g} {5} <a<g\)
4.
\(a = g \)
Subtopic: Â Circular Motion |
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Two particles \(A\), \(B\) are projected simultaneously from the base of a triangle \(ABC\). Particle \(A\) is projected from vertex \(A\) along \(AC,\) and particle \(B\) is projected from vertex \(B\) along \(BC\). Their respective velocities are \(v_A\) & \(v_B\) and they move with uniform velocities. For the particles to collide:
| 1. | \(v_A~\text{cos}A=v_B~\text{cos}B\) |
| 2. | \(v_A~\text{sin}A=v_B~\text{sin}B\) |
| 3. | \(\dfrac{v_A}{\text{sin}A}=\dfrac{v_B}{\text{sin}B}\) |
| 4. | \(v_A~\text{tan}A=v_B~\text{tan}B\) |
Subtopic: Â Relative Motion |
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Particles are simultaneously projected in all possible directions from a point in space, located in a uniform gravitational field. The initial speed of the particle is \(u.\) The maximum separation between any two particles, after a time \(t,\) is:
| 1. | \(ut\) | 2. | \(2ut\) |
| 3. | \(ut+\dfrac{1}{2}gt^2\) | 4. | \(2ut+gt^2\) |
Subtopic: Â Projectile Motion |
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A boat is rowed across a \(400~\text m\) wide river so that it can reach the opposite bank in a minimum of \(10\) minutes. No matter which direction the boat is rowed in, it cannot reach a point exactly opposite on the other bank, unless it is rowed at a slightly higher speed. The speed of flow of the river is:
| 1. | \(2.4~\text{km/h}\) | 2. | \(4.8~\text{km/h}\) |
| 3. | \(2.4\sqrt2~\text{km/h}\) | 4. | \(\dfrac{2.4}{\sqrt2}~\text{km/h}\) |
Subtopic: Â Relative Motion |
 59%
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Raindrops fall from the sky making an angle of \(30^\circ\) with the vertical. If a man runs at \(2\) m/s, he finds that the drops fall vertically. If he were to run in the opposite direction with the same speed\(,\) the raindrops will fall with a vertical speed of:
| 1. | \(2\) m/s | 2. | \(4\) m/s |
| 3. | \(2\sqrt3 \) m/s | 4. | \(4\sqrt3 \) m/s |
Subtopic: Â Relative Motion |
 63%
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Wind blows towards north with a speed of \(20\) m/s. A plane flies in air with a velocity relative to ground of \(100\) m/s, towards east. Then, the velocity of the plane relative to air is:
| 1. | \(100\) m/s towards east and \(20\) m/s towards north. |
| 2. | \(100\) m/s towards east and \(20\) m/s towards south. |
| 3. | \(100\) m/s towards west and \(20\) m/s towards north. |
| 4. | \(100\) m/s towards west and \(20\) m/s towards south. |
Subtopic: Â Relative Motion |
 69%
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Consider the two situations shown in the figures. In figure (A), the particle \(P\) is given a velocity \(u\) up a smooth horizontal incline and it reaches a maximum vertical height \(h_A\). In figure (B), the same particle \(P\) is projected with a velocity \(u\) at an angle \(\theta\) (parallel to the previous incline) and reaches a maximum height \(h_B\).Then,
| 1. | \(h_A=h_B~\text{sin}\theta\) |
| 2. | \(h_A~\text{sin}\theta=h_B\) |
| 3. | \(h_A~\text{sin}^2\theta=h_B\) |
| 4. | \(\dfrac{h_A}{\text{sin}^2\theta}=h_B\) |
Subtopic: Â Projectile Motion |
 53%
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The average velocity of a projectile from the point of projection to impact is \(v_1\) while the average velocity from projection to maximum height\((H)\) is \(v_2\).
It can be concluded that:
It can be concluded that:
| 1. | \(v_1>v_2\) |
| 2. | \(v_1<v_2\) |
| 3. | \(v_1=v_2\) |
| 4. | Any of the above can be true depending on the angle of projection |
Subtopic: Â Projectile Motion |
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A man, swimming with a speed \(u_1,\) can cross a river fastest in a time, \(T\). His friend, who swims with a speed \(u_2,\) reaches the opposite bank in the same time when he swims at an angle of \(30^{\circ}\) with the bank. Then:
| 1. | \(u_1=\dfrac{\sqrt3}{2}u_2\) | 2. | \(u_1=\dfrac{1}{2}u_2\) |
| 3. | \(u_1=\dfrac{1}{\sqrt2}u_2\) | 4. | \(u_1=\dfrac{1}{\sqrt3}u_2\) |
Subtopic: Â Relative Motion |
 58%
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A ball is thrown up with a speed \(u\) at an angle of \(60^{\circ}\) with the horizontal; the thrower of the ball runs with a uniform speed \(v\) and stops suddenly when he reaches a certain point. He observes that the ball is at its maximum height, and then waits until it reaches him. Then
1. \(v=4u\)
2. \(v=2u\)
3. \(v=u\)
4. \(v<u\)
1. \(v=4u\)
2. \(v=2u\)
3. \(v=u\)
4. \(v<u\)
Subtopic: Â Projectile Motion |
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