A particle starts from the origin at \(t=0\) and moves in the \(x\text-y\) plane with constant acceleration \(a\) in the \(y\text-\)direction. Its equation of motion is, \(y=bx^{2}.\) The \(x\) component of its velocity is:
1. variable
2. \(\sqrt{\frac{2a}{b}}\)
3. \(\frac{a}{2b}\)
4. \(\sqrt{\frac{a}{2b}}\)
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 \(r\). |
A particle has an initial velocity (\(2\hat{i}+3\hat{j}\)) and an acceleration (\(0.3\hat{i}+0.2\hat{j}\)). The magnitude of velocity after \(10\) s will be:
1. \(9 \sqrt{2} ~\text{units} \)
2. \(5 \sqrt{2} ~\text{units} \)
3. \(5 ~\text{units} \)
4. \(9~\text{units} \)
A particle moves in the x-y plane according to rule and . The particle follows:
1. | an elliptical path. |
2. | a circular path. |
3. | a parabolic path. |
4. | a straight line path inclined equally to the x and y-axis. |
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 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. \(50~\text{m}\)
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. | \(R=\frac{H_1{ }^2}{H_2{ }^2}\) |
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. | \(S\) |
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. |
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. | \(t^{2} \sqrt{\alpha^{2} +\beta^{2}}\) |