Q. No.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 \(r.\) |
The \(x\) and \(y\) coordinates of the particle at any time are \(x=5 t-2 t^2\) and \({y}=10{t}\) respectively, where \(x\) and \(y\) are in meters and \(\mathrm{t}\) in seconds. The acceleration of the particle at \(\mathrm{t}=2\) s is:
| 1. | \(5\hat{i}~\text{m/s}^2\) | 2. | \(-4\hat{i}~\text{m/s}^2\) |
| 3. | \(-8\hat{j}~\text{m/s}^2\) | 4. | \(0\) |
A particle moves in space such that:
\(x=2t^3+3t+4;~y=t^2+4t-1;~z=2\sin\pi t\)
where \(x,~y,~z\) are measured in meters and \(t\) in seconds. The acceleration of the particle at \(t=3\) seconds will be:
| 1. | \(36 \hat{i}+2 \hat{j}+\hat{k} \) ms-2 |
| 2. | \(36 \hat{i}+2 \hat{j}+\pi \hat{k} \) ms-2 |
| 3. | \(36 \hat{i}+2 \hat{j} \) ms-2 |
| 4. | \(12 \hat{i}+2 \hat{j} \) ms-2 |
A body is moving with a velocity of \(30~\text{m/s}\) towards the east. After \(10~\text s,\) its velocity becomes \(40~\text{m/s}\) towards the north. The average acceleration of the body is:
1. \( 7~\text{m/s}^2\)
2. \( \sqrt{7}~\text{m/s}^2\)
3. \(5~\text{m/s}^2\)
4. \(1~\text{m/s}^2\)
A particle moves along a parabolic path \(y =9x^2\) in such a way that the \(x\) component of the velocity remains constant and has a value of \(\frac{1}{3}~\text{m/s}\). It can be deduced that the acceleration of the particle will be:
1. \(\frac{1}{3}\hat j~\text{m/s}^2\)
2. \(3\hat j~\text{m/s}^2\)
3. \(\frac{2}{3}\hat j~\text{m/s}^2\)
4. \(2\hat j~\text{m/s}^2\)
A particle starts from the origin at t=0 and moves in the x-y plane with a constant acceleration 'a' in the y direction. Its equation of motion is . The x component of its velocity (at t=0) will be:
1. variable
2.
3.
4.
A particle is moving eastwards with velocity of \(5\) m/s. In \(10\) seconds the velocity changes to \(5\) m/s northwards. The average acceleration in this time is?
| 1. | zero |
| 2. | \(\frac{1}{\sqrt{2}}~ \text{m/s}^2\) toward north-west |
| 3. | \(\frac{1}{\sqrt{2}}~\text{m/s}^2\) toward north-east |
| 4. | \(\frac{1}{2}~\text{m/s}^2 \) toward north-west |
If the position of a particle varies according to the equations \(x= 3t^2\), \(y =2t\), and \(z= 4t+4\), then which of the following is incorrect?
| 1. | Velocities in \(y\) and \(z\) directions are constant |
| 2. | Acceleration in the \(x\text-\)direction is non-uniform |
| 3. | Acceleration in the \(x\text-\)direction is uniform |
| 4. | Motion is not in a straight line |
A particle moving on a curved path possesses a velocity of \(3\) m/s towards the north at an instant. After \(10\) s, it is moving with speed \(4\) m/s towards the west. The average acceleration of the particle is:
| 1. | \(0.25~\text{m/s}^2,\) \(37^{\circ}\) south to east. |
| 2. | \(0.25~\text{m/s}^2,\) \(37^{\circ}\) west to north. |
| 3. | \(0.5~\text{m/s}^2,\) \(37^{\circ}\) east to north. |
| 4. | \(0.5~\text{m/s}^2,\) \(37^{\circ}\) south to west. |
The position vector of a particle \(\overrightarrow r\) as a function of time \(t\) (in seconds) is \(\overrightarrow r=3 t \hat{i}+2t^2\hat j~\text{m}\). The initial acceleration of the particle is:
1. \(2~\text{m/s}^2\)
2. \(3~\text{m/s}^2\)
3. \(4~\text{m/s}^2\)
4. zero
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