Q. No.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 \(\left(\frac{\pi}{2}-\theta\right)\) with the horizontal, it reaches maximum height \(H_2.\) The relation between the horizontal range of the projectile \(R\) and \(H_1\) and \(H_2\) 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 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. \(5\sqrt{2}\)
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 a 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. | \(t^{2} \sqrt{\alpha^{2} +\beta^{2}}\) |
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. | \(6 \hat{i}+2 \hat{j}-3 \hat{k} \) |
| 2. | \(-18 \hat{i}-13 \hat{j}+2 \hat{k} \) |
| 3. | \(4 \hat{i}-13 \hat{j}+6 \hat{k}\) |
| 4. | \(6 \hat{i}-2 \hat{j}+8 \hat{k}\) |
The time taken by a block of wood (initially at rest) to slide down a smooth inclined plane \(9.8~\text{m}\) long (angle of inclination is \(30^{\circ}\)
| 1. | \(\frac{1}{2}~\text{sec} \) | 2. | \(2 ~\text{sec} \) |
| 3. | \(4~ \text{sec} \) | 4. | \(1~\text{sec} \) |
An airplane is moving with a velocity \(u.\) It drops a packet from a height \(h.\) The time \(t\) taken by the packet to reach the ground will be:
1. \( \sqrt{\frac{2 g}{h}} \)
2. \( \sqrt{\frac{2 u}{g}} \)
3. \( \sqrt{\frac{h}{2 g}} \)
4. \( \sqrt{\frac{2 h}{g}}\)
A particle moves so that its position vector is given by \(r=\cos \omega t \hat{x}+\sin \omega t \hat{y}\) where \(\omega\) is a constant. Based on the information given, which of the following is true?
| 1. | The velocity and acceleration, both are parallel to \(r.\) |
| 2. | The velocity is perpendicular to \(r\) and acceleration is directed towards the origin. |
| 3. | The velocity is not perpendicular to \(r\) and acceleration is directed away from the origin. |
| 4. | The velocity and acceleration, both are perpendicular to \(r.\) |
Two men \(P\) and \(Q\) are standing at corners \(A\) and \(B\) of a square \(ABCD\) of side \(8~\text m.\) They start moving along the track with a constant speed \(2~\text{m/s}\) and \(10~\text {m/s}\) respectively. The time when they will meet for the first time is equal to:
1. \(2~\text{sec}\)
2. \(3~\text{sec}\)
3. \(1~\text{sec}\)
4. \(6~\text{sec}\)
The velocity of a projectile at the initial point \(A\) is \(2\hat i+3\hat j~\text{m/s}.\) Its velocity (in m/s) at the point \(B\) is:
| 1. | \(-2\hat i+3\hat j~\) | 2. | \(2\hat i-3\hat j~\) |
| 3. | \(2\hat i+3\hat j~\) | 4. | \(-2\hat i-3\hat j~\) |
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