Q. No.A particle moves with constant speed \(v\) along a circular path of radius \(r\) and completes the circle in time \(T\). The acceleration of the particle is:
1. \(2\pi v / T\)
2. \(2\pi r / T\)
3. \(2\pi r^2 / T\)
4. \(2\pi v^2 / T\)
1. \(2\pi v / T\)
2. \(2\pi r / T\)
3. \(2\pi r^2 / T\)
4. \(2\pi v^2 / T\)
If the equation for the displacement of a particle moving on a circular path is given by \(\theta = 2t^3 + 0.5\) where \(\theta\) is in radians and \(t\) in seconds, then the angular velocity of the particle after \(2\) sec from its start is:
1. \(8\) rad/sec
2. \(12\) rad/sec
3. \(24\) rad/sec
4. \(36\) rad/sec
The coordinates of a moving particle at any time \(t\) are given by \(x= \alpha t^3\) and \(y = \beta t^3\). The speed of the particle at time \(t\) is given by:
| 1. | \(\sqrt{\alpha^{2} + \beta^{2}}\) | 2. | \(3t \sqrt{\alpha^{2} + \beta^{2}}\) |
| 3. | \(3t^{2} \sqrt{\alpha^{2} +\beta^{2}}\) | 4. | \(t^{2} \sqrt{\alpha^{2} +\beta^{2}}\) |
At the top of the trajectory of a projectile, the directions of its velocity and acceleration are:
| 1. | perpendicular to each other |
| 2. | parallel to each other |
| 3. | inclined to each other at an angle of \(45^\circ\) |
| 4. | antiparallel to each other |
Four bodies \(P\), \(Q\), \(R\) and \(S\) are projected with equal velocities having angles of projection \(15^{\circ},\) \(30^{\circ},\)\(45^{\circ},\) and \(60^{\circ}\) with the horizontal respectively. The body having the shortest range is?
| 1. | \(P\) | 2. | \(Q\) |
| 3. | \(R\) | 4. | \(S\) |
A stone projected with a velocity \(u\) at an angle \(\theta\) with the horizontal reaches maximum height \(H_1\). When it is projected with velocity \(u\) at an angle \(\frac{\pi}{2}-\theta\) with the horizontal, it reaches maximum height \(H_2\). The relation between the horizontal range of the projectile \(R\) and \(H_1\) & \(H_2\) is:
| 1. | \(R=4 \sqrt{H_1 H_2} \) | 2. | \(R=4\left(H_1-H_2\right) \) |
| 3. | \(R=4\left(H_1+H_2\right) \) | 4. | \(R=\frac{H_1{ }^2}{H_2{ }^2}\) |
A cricketer can throw a ball to a maximum horizontal distance of \(100~\text{m}\). With the same effort, he throws the ball vertically upwards. The maximum height attained by the ball is:
1. \(100~\text{m}\)
2. \(80~\text{m}\)
3. \(60~\text{m}\)
4. \(50~\text{m}\)
The horizontal range of a projectile is \(4 \sqrt{3}\) times its maximum height. Its angle of projection will be:
1. \(45^{\circ}\)
2. \(60^{\circ}\)
3. \(90^{\circ}\)
4. \(30^{\circ}\)
A boat is moving with a velocity \(3\hat i + 4\hat j\) with respect to ground. The water in the river is moving with a velocity\(-3\hat i - 4 \hat j\) with respect to ground. The relative velocity of the boat with respect to water is:
1. \(8\hat j\)
2. \(-6\hat i-8\hat j\)
3. \(6\hat i+8\hat j\)
4. \(5\sqrt{2}\)
A river is flowing from east to west at a speed of \(5\) m/min. A man on south bank of river, capable of swimming \(10\) m/min in still water, wants to swim across the river in shortest time. He should swim:
| 1. | Due north |
| 2. | Due north-east |
| 3. | Due north-east with double the speed of the river |
| 4. | None of the above |
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