A cat is situated at point \(A\) (\(0,3,4\)) and a rat is situated at point \(B\) (\(5,3,-8\)$\mathrm{}$). The cat is free to move but the rat is always at rest. The minimum distance travelled by the cat to catch the rat is:
1. 5 unit
2. 12 unit
3. 13 unit
4. 17 unit

Balls A and B are thrown from two points lying on the same horizontal plane separated by a distance of 120 m. Which of the following statements is correct?

1. The balls can never meet.

2. The balls can meet if the ball B is thrown 1 s later.

3. The two balls meet at a height of 45 m.

4. None of the above

A body is projected at an angle of $30°$ with the horizontal with a speed of 30 m/s. What is the angle made by the velocity vector with the horizontal after 1.5 sec? (g = 10$\mathrm{m}/{\mathrm{s}}^2$)

1. $0°$

2. $30°$

3. $60°$

4. $90°$

A particle moves along the positive branch of the curve \(y= \frac{x^{2}}{2}\) where \(x= \frac{t^{2}}{2}\), & \(x\) and \(y\) are measured in metres and in seconds respectively. At \(t= 2~\text{s}\), the velocity of the particle will be:
1. \(\left(\right. 2 \hat{i} - 4 \hat{j})~\text{m/s}\) 
2. \(\left(\right. 4 \hat{i} + 2 \hat{j}\left.\right)\text{m/s}\)
3. \(\left(\right. 2 \hat{i} + 4 \hat{j}\left.\right) \text{m/s}\)
4. \(\left(\right. 4 \hat{i} - 2 \hat{j}\left.\right) \text{m/s}\)

A deer wants to save her life from a lion. The lion follows a path whose equation is ${\mathrm{x}}^2+{\mathrm{y}}^2=16$. For saving her life, the deer needs to move on a path whose equation will be:

1. ${\mathrm{x}}^2+{\mathrm{y}}^2=4$

2. ${\mathrm{x}}^2+{\mathrm{y}}^2=16$

3. ${\mathrm{x}}^2+{\mathrm{y}}^2-64=0$

4. Both (1) and (3) are correct

A particle rotates in a circular path starting from rest. If angular acceleration is 4 $rad/s^2$, then the time after which angle between net acceleration and tangential acceleration becomes $45°\; is?$

The position of a moving particle at time \(t\) is \(\vec{r}=3\hat{i}+4t^{2}\hat{j}-t^{3}\hat{k}.\) Its displacement during the time interval \(t=1\) s to \(t=3\) s will be:

An object moves at a constant speed along a circular path in a horizontal XY plane with its centre at the origin. When the object is at x = –2 m, its velocity is –(4 m/s)$\hat{\mathrm{j}}$. What is the object's acceleration when it is at y = 2 m?

1. $-(8\mathrm{ m}/{\mathrm{s}}^2)$ $\hat{\mathrm{j}}$

2. $-(8\mathrm{ m}/{\mathrm{s}}^2)$ $\hat{\mathrm{i}}$

3. $-(4\mathrm{ m}/{\mathrm{s}}^2)$ $\hat{\mathrm{j}}$

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

A particle has an initial velocity $\overrightarrow{\mathrm{u}}=(4\hat{i}-5\hat{j})$ m/s and it is moving with an acceleration $\overrightarrow{\mathrm{a}}=\left(\frac{1}{4}\hat{i}+\frac{1}{5}\hat{j}\right)m/s^2$. Velocity of the particle at \(t=2\) s will be:

1. $(6\hat{\mathrm{i}}-4\hat{\mathrm{j}})\mathrm{m}/\mathrm{s}$

2. $(4.5\hat{i}-4.5\hat{\mathrm{j}})\mathrm{m}/\mathrm{s}$

3. $(4.5\hat{i}-4.6\hat{\mathrm{j}})\mathrm{m}/\mathrm{s}$

4. $(6\hat{\mathrm{i}}-4.6\hat{\mathrm{j}})\mathrm{m}/\mathrm{s}$

To the captain of a ship A travelling with velocity $\overrightarrow{v_A}=(3\hat{i}-4\hat{j})$ km/h, a second ship B appears to have a velocity ${\overrightarrow{v}}_B=\left(5\hat{i}+12\hat{j}\right)$ km/h. What is the true velocity of the ship B?

1. $2\hat{i}+16\hat{j}$ $km/h$

2. $13\hat{i}+8\hat{j}$ $km/h$

3. $-2\hat{i}-16\hat{j}$ $km/h$

4. none of these