Particles P and Q are at ${\overrightarrow{\mathrm{r}}}_{\mathrm{P}} = 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}}$ and ${\overrightarrow{\mathrm{r}}}_{\mathrm{Q}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}}$. Their velocities at these positions are ${\overrightarrow{\mathrm{v}}}_{\mathrm{P}} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}}$ and ${\overrightarrow{\mathrm{v}}}_{\mathrm{Q}} = 3\hat{\mathrm{i}} + \mathrm{c}\hat{\mathrm{j}}$ respectively. If they collide after one second, then the value of c is

(1) 1

(2) 2

(3) 3

(4) 4

Two projectiles are thrown at different angles but their ranges R are the same. If their times of flight are ${\mathrm{t}}_1 \mathrm{and} {\mathrm{t}}_2$, then the product of their times of flight ${\mathrm{t}}_1{\mathrm{t}}_2$ is proportional to

1.  $\frac{1}{\mathrm{R}}$

2.  $\frac{1}{{\mathrm{R}}^2}$

3.  ${\mathrm{R}}^2$

4.  R

It is raining at \(20\) m/s in still air. Now a wind starts blowing with speed \(10\) m/s in the north direction. If a cyclist starts moving at \(10\) m/s in the south direction, then the apparent velocity of rain with respect to a cyclist will be:
1. \(20\) m/s
2. \(20\sqrt{2}\) m/s
3. \(10 \sqrt{5}\) m/s
4. \(30\) m/s

A body of mass 2 kg is projected at a speed of 20 m/s from a pillar of height 40 m at an angle of 30° with the horizontal in an upward direction. The speed with which it will hit the ground is

(1) 30 m/s

(2) 20$\sqrt{3}$ m/s

(3) 25 m/s

(4) 40 m/s

Path of a projectile with respect to another projectile so long as both remain in the air is:
1. Circular
2. Parabolic
3. Straight
4. Hyperbolic

A particle is thrown with velocity u making an angle $\mathrm{\theta }$ with the horizontal. It just crosses the top of two poles each of height h after 1 s and 3 s respectively. The time of flight of the particle is:

(1) 4 s

(2) 2 s

(3) 8 s

(4) 6 s

A projectile is moving in the XY plane such that the horizontal and vertical directions are along the X-axis and Y-axis respectively. The projectile is fired from the origin at a large height with initial velocity ($50\hat{\mathrm{i}} + 20\hat{\mathrm{j}}$) m/s. The time after which instantaneous velocity of the projectile becomes perpendicular to its initial velocity is:     (g=10m/s²)

(1) 7.5 s

(2) 14.5 s

(3) 5.5 s

(4) 10.5 s

A particle is thrown from the ground with a speed u at an angle $\mathrm{\theta }$ above the horizontal. The rate of change of velocity of the particle at the highest point of the path is:

1. gsin$\mathrm{\theta }$

2. gcos$\mathrm{\theta }$

3. g

4. Zero

A missile is fired for maximum range with an initial velocity of \(20\) m/s, then the maximum height of missile is: (Take \(g=10\) m/s²)
1. \(20\) m
2. \(30\) m
3. \(10\) m
4. \(40\) m

A particle is moving along a circle of radius \(R \) with constant speed \(v_0\). What is the magnitude of change in velocity when the particle goes from point \(A\) to \(B \) as shown?