Q. No.The position vector of a particle as a function of time t (in seconds) is . The initial acceleration of the particle is:
1. 2
2. 3
3. 4
4. Zero
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 body is moving with a velocity of \(30\) m/s towards the east. After \(10\) s, its velocity becomes \(40\) m/s towards the north. The average acceleration of the body is:
1. \( 7 \mathrm{~m} / \mathrm{s}^2 \)
2. \( \sqrt{7} \mathrm{~m} / \mathrm{s}^2 \)
3. \( 5 \mathrm{~m} / \mathrm{s}^2 \)
4. \( 1 \mathrm{~m} / \mathrm{s}^2\)
If the position of a particle varies according to the equations x = 3, 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-direction is non-uniform |
| 3. | Acceleration in the x-direction is uniform |
| 4. | Motion is not in a straight line |
A particle is moving in the XY plane such that \(x = \left(t^2 -2t\right)\) m, and \(y = \left(2t^2-t\right)\) m, then:
| 1. | Acceleration is zero at t = 1 sec |
| 2. | Speed is zero at t = 0 sec |
| 3. | Acceleration is always zero |
| 4. | Speed is 3 m/s at t = 1 sec |
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 moves in space such that:
\(x=2t^3+3t+4;~y=t^2+4t-1;~z=2sin\pi t\)
where \(x,~y,~z\) are measured in meters and \(t\) in seconds. The acceleration of the particle at \(t=3\) seconds will be:
| 1. | \(36 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}} \) ms-2 |
| 2. | \(36 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\pi \hat{\mathrm{k}} \) ms-2 |
| 3. | \(36 \hat{\mathrm{i}}+2 \hat{\mathrm{j}} \) ms-2 |
| 4. | \(12 \hat{\mathrm{i}}+2 \hat{\mathrm{j}} \) ms-2 |
A particle is moving along a curve. Select the correct statement.
| 1. | If its speed is constant, then it has no acceleration. |
| 2. | If its speed is increasing, then the acceleration of the particle is along its direction of motion. |
| 3. | If its speed is decreasing, then the acceleration of the particle is opposite to its direction of motion. |
| 4. | If its speed is constant, its acceleration is perpendicular to its velocity. |
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 , 37° south to east |
| 2. | 0.25 , 37° west to north |
| 3. | 0.5 , 37° east to north |
| 4. | 0.5 , 37° south to west |
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. | \(\frac{1}{\sqrt{2}} \mathrm{~m} / \mathrm{s}^2 \) toward north-west |
| 3. | \(\frac{1}{\sqrt{2}} \mathrm{~m} / \mathrm{s}^2 \) toward north-east |
| 4. | \(\frac{1}{2} m / s^2 \) toward north-west |
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