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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 $\mathrm{}$\(30^{\circ}\)

** **

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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