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 (\(2\mathrm{i}+3\mathrm{j}\)) and an acceleration (\(0.3\mathrm{i}+0.2\mathrm{j}\)). 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. |
A particle starts from the origin at \(t=0\) and moves in the \(x -y\) plane with constant acceleration \(a\) in the \(y-\)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. \(\sqrt{\frac{a}{2b}}\)
The horizontal range of a projectile is times its maximum height. Its angle of projection will be:
1. 45°
2. 60°
3. 90°
4. 30°
A cricketer can throw a ball to a maximum horizontal distance of 100 m. With the same effort, he throws the ball vertically upwards. The maximum height attained by the ball is
1. 100 m
2. 80 m
3. 60 m
4. 50 m
A stone projected with a velocity u at an angle θ with the horizontal reaches maximum height H1. When it is projected with velocity u at an angle with the horizontal, it reaches maximum height H2. The relation between the horizontal range of the projectile R and H1 & H2 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 15o, 30o, 45o and 60o with the horizontal respectively. The body having the shortest range is?
1. | P | 2. | Q |
3. | R | 4. | S |
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 |
The coordinates of a moving particle at any time ‘t’ are given by x = αt3 and y = βt3. The speed of the particle at time ‘t’ is given by:
1.
2.
3.
4.