The position vector of a particle $\overrightarrow{\mathrm{r}}$ as a function of time t (in seconds) is $\overrightarrow{\mathrm{r}} = \left(3\mathrm{t}\right)\hat{\mathrm{i}} + \left(2{\mathrm{t}}^2\right)\hat{\mathrm{j}} \mathrm{m}$. The initial acceleration of the particle is:

1. 2 $\mathrm{m}/{\mathrm{s}}^2$

2. 3 $\mathrm{m}/{\mathrm{s}}^2$

3. 4 $\mathrm{m}/{\mathrm{s}}^2$

4. Zero

The x and y coordinates of the particle at any time are $\mathrm{x}= 5\mathrm{t}-2{\mathrm{t}}^2$ and y= 10t respectively, where x and y are in meters and t in seconds. The acceleration of the particle at t= 2sec is:

1. $5\hat{i} \mathrm{m}/{\mathrm{s}}^2$

2. $-4\hat{i} \mathrm{m}/{\mathrm{s}}^2$

3. $-8\hat{j} \mathrm{m}/{\mathrm{s}}^2$

4. 0

A body is moving with a velocity of 30 m/s towards the east. After 10 sec, 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 position of a particle varies according to the equations x = 3${\mathrm{t}}^2$, 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-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 = $\left({\mathrm{t}}^2 - 2\mathrm{t}\right)$ m, and $\mathrm{y} = \left(2{\mathrm{t}}^2 - \mathrm{t}\right)$ 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 $r=cos\omega t \hat{x}+sin\omega t \widehat{y }where \omega$ 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

$\mathrm{x}=2{\mathrm{t}}^3+3\mathrm{t}+4$

$\mathrm{y}={\mathrm{t}}^2+4\mathrm{t}-1$

$\mathrm{z}=2 \sin \mathrm{\pi t}$

where x, y, z are measured in metres and t in seconds. The acceleration of the particle at t=3s will be

1.  $36\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}} {\mathrm{ms}}^{-2}$

2.  $36\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\mathrm{\pi }\hat{\mathrm{k}} {\mathrm{ms}}^{-2}$

3.  $36\hat{\mathrm{i}}+2\hat{\mathrm{j}} {\mathrm{ms}}^{-2}$

4.  $12\hat{\mathrm{i}}+2\hat{\mathrm{j}} {\mathrm{ms}}^{-2}$

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 $\mathrm{m}/{\mathrm{s}}^2$, 37° south to east

2. 0.25 $\mathrm{m}/{\mathrm{s}}^2$, 37° west to north

3. 0.5 $\mathrm{m}/{\mathrm{s}}^2$, 37° east to north

4. 0.5 $\mathrm{m}/{\mathrm{s}}^2$, 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. $\frac{1}{\sqrt{2}}m\text{/}{s}^{\text{2}}$ toward north-west

3. $\frac{1}{\sqrt{2}}m\text{/}{s}^{\text{2}}$ toward north-east

4. $\frac{1}{2}m\text{/}{s}^{\text{2}}$ toward north-west