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. $$t^{2} \sqrt{\alpha^{2} +\beta^{2}}$$

Subtopic:  Speed & Velocity |
79%
From NCERT
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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 $$R=\frac{H_1{ }^2}{H_2{ }^2}$$
Subtopic:  Projectile Motion |
83%
From NCERT
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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. $$5\sqrt{2}$$

Subtopic:  Relative Motion |
83%
From NCERT
PMT - 1998
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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. $$2\pi v^2 / T$$

Subtopic:  Circular Motion |
62%
From NCERT
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What is the value of linear velocity if $$\overrightarrow{\omega} = 3\hat{i} - 4\hat{j} + \hat{k}$$ and $$\overrightarrow{r} = 5\hat{i} - 6\hat{j} + 6\hat{k}$$:

 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 $$6 \hat{i}-2 \hat{j}+8 \hat{k}$$
Subtopic:  Circular Motion |
86%
From NCERT
PMT - 2000
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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 $$30^{\circ}$$) is:

 1 $$\frac{1}{2}~\text{sec}$$ 2 $$2 ~\text{sec}$$ 3 $$4~ \text{sec}$$ 4 $$1~\text{sec}$$
Subtopic:  Uniformly Accelerated Motion |
71%
From NCERT
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An aeroplane 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. $$\sqrt{\left(\frac{2 g}{h}\right)}$$
2. $$\sqrt{\left(\frac{2 u}{g}\right)}$$
3. $$\sqrt{\left(\frac{h}{2 g}\right)}$$
4. $$\sqrt{\left(\frac{2 h}{g}\right)}$$

Subtopic:  Projectile Motion |
84%
From NCERT
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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 Velocity and acceleration, both are parallel to $$r$$. 2 Velocity is perpendicular to $$r$$ and acceleration is directed towards the origin. 3 Velocity is not perpendicular to $$r$$ and acceleration is directed away from the origin. 4 Velocity and acceleration, both are perpendicular to $$r$$.
Subtopic:  Acceleration |
70%
From NCERT
NEET - 2016
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Two men $$P$$ & $$Q$$ are standing at corners $$A$$ & $$B$$ of square $$ABCD$$ of side $$8$$ m. They start moving along the track with constant speed $$2$$ m/s and $$10$$ m/s respectively. The time when they will meet for the first time, is equal to:

1. $$2$$ sec
2. $$3$$ sec
3. $$1$$ sec
4. $$6$$ sec

Subtopic:  Relative Motion |
56%
From NCERT
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The velocity of a projectile at the initial point $$A$$ is $$2\hat i+3\hat j~$$m/s. Its velocity (in m/s) at 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 $$-2\hat i-3\hat j~$$
Subtopic:  Projectile Motion |
83%
From NCERT
AIPMT - 2013
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