Two particles having position vectors \(\overrightarrow{r_{1}} = \left( 3 \hat{i} + 5 \hat{j}\right)\) metres and \(\overrightarrow{r_{2}} = \left(- 5 \hat{i} - 3 \hat{j} \right)\) metres are moving with velocities \(\overrightarrow{v}_{1} = \left( 4 \hat{i} + 3 \hat{j}\right)\)\(\text{m/s}\) and \(\overrightarrow{v}_{2} = \left(\alpha\hat{i} + 7 \hat{j} \right)\)\(\text{m/s}\). If they collide after \(2\) seconds, the value of \(\alpha\) is:

1.	\(2\)	2.	\(4\)
3.	\(6\)	4.	\(8\)

A batsman hits a sixer and the ball touches the ground outside the cricket field. Which of the following graph describes the variation of the cricket ball's vertical velocity \(v\) with time between the time \(t_1\) as it hits the bat and time \(t_2\) when it touches the ground:

1.		2.
3.		4.

The \(x\) and \(y\) coordinates of the particle at any time are \(x = 5t-2t^2\) and \(y=10t\) respectively, where \(x\) and \(y\) are in metres and \(t\) is in seconds. The acceleration of the particle at \(t=2\) s is:

Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time $t_1$. On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time $t_2$. The time taken by her to walk up on the moving escalator will be 

(1)$\frac{t_1-t_2}{2}$

(2)$\frac{t_1{t}_2}{t_2-t_1}$

(3) $\frac{t_1{t}_2}{t_2+t_1}$

(4) $t_1-t_2$

In the given figure, $a=15 m/s^2$ represents the total acceleration of a particle moving in the clockwise direction in a circle of radius R=2.5 m at a given instant of time. The speed of the particle is:

1. 4.5 m/s

2. 5.0 m/s

3. 5.7 m/s 

4. 6.2 m/s

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?

A particle of mass 10g moves along a circle of radius 6.4 cm with a constant tangential acceleration. What is the magnitude of this acceleration, if the kinetic energy of the particle becomes equal to 8x10^-4 J by the end of the second revolution after the beginning of the motion?

(1) 0.15 m/s²

(2) 0.18 m/s²

(3) 0.2 m/s²

(4) 0.1 m/s²

A ship A is moving Westwards with a speed of 10 km h^-1 and a ship B 100 km South of A, is moving Northwards with a speed of 10 km h^-1. The time after which the distance between them becomes shortest is

1., h
2. 5h
3. 5$\sqrt{2}$h
4. 10$\sqrt{2}$h

A projectile is fired from the surface of the earth with a velocity of 5 m/s and angle $\mathrm{\theta }$ with the horizontal. Another projectile fired from another planet with a velocity of 3 m/s at the same angle follows a trajectory, which is identical to the trajectory of the projectile fired from the earth. The value of the acceleration due to gravity on the planet (in m/s²) is: [Given, g = 9.8 m/s²]

1. 3.5

2. 5.9

3. 16.3

4. 110.8

A particle is moving such that its position coordinates (x, y) are (2m, 3m) at time t = 0, (6m, 7m) at time t = 2s and (13m, 14m) at time t = 5s. Average velocity vector (v_av) from t = 0 to t = 5s is

1. $\frac{1}{5}$(13$\hat{i}$+14$\hat{j}$)

2. $\frac{7}{3}$($\hat{i}$+$\hat{j}$)

3. 2($\hat{i}$+$\hat{j}$)

4. $\frac{11}{5}$($\hat{i}$+$\hat{j}$)