Raindrops are falling with a speed v vertically downwards and a man is running on a horizontal road with speed u. The magnitude of the velocity of the raindrops with respect to the man is

A bomb is dropped from an aeroplane flying horizontally. The path of the bomb as seen by the pilot will be (neglect air friction)

The range of a bullet fired from a gun at an angle $\left(\frac{\mathrm{\pi }}{4} - \mathrm{\varphi }\right)$ with the horizontal is the same as the range of another bullet fired from that gun at an angle $\theta$ with the horizontal. Then,

A boy runs on a circular track of radius R (in km) with a speed of $\frac{\mathrm{\pi R}}{2}$ km/h in the clockwise direction for 3 h and then with $\mathrm{\pi R}$ km/h in the anticlockwise direction for 1 h. The magnitude of his displacement will be 

Two projectiles P and Q are projected with the same speed at angles 60° and 30° with horizontal on level ground, then: [R = Range, T = Time of flight, H = Maximum height]

A body is projected with the same speed at two different angles covers the same horizontal distance R. If ${\mathrm{T}}_1 \mathrm{and} {\mathrm{T}}_2$ are two times of flights, then R is equal to :

A body starts moving from rest on a horizontal ground such that the position vector of the body with respect to its starting point is given by $\mathrm{r} = 2\mathrm{t}\hat{\mathrm{i}} + 3{\mathrm{t}}^2\hat{\mathrm{j}}$. The equation of the trajectory of the body is:

A particle starts moving with constant acceleration with initial velocity ($\hat{\mathrm{i}} + 5\hat{\mathrm{j}}$)m/s. After 4 seconds, its velocity becomes ($3\hat{\mathrm{i}} - 2\hat{\mathrm{j}}$) m/s. The magnitude of its displacement in 4 seconds is:

A particle is revolving in a circular path of radius 200 m at a speed of 20 m/s. Its speed is increasing at a rate of $\sqrt{5}$ $\mathrm{m}/{\mathrm{s}}^2$. Its acceleration is :

The ratio of maximum height attained by two projectiles of the same mass is 25: 9. The ratio of their minimum kinetic energy is 4: 9. The ratio of their ranges is:

If a particle is moving in a circular orbit with constant speed, then

In a uniform circular motion,

The relative velocity of a car P with respect to Q is $20\sqrt{2}$ m/s North-East. The velocity of car Q is 40 m/s due to South. The velocity of car P is:

A projectile is projected with initial kinetic energy K. If it has kinetic energy 0.25K at its highest point, then the angle of projection is:

A body is projected from the ground at an angle of 30° with horizontal. The instantaneous velocity of the body becomes perpendicular to initial velocity during its motion:

A body is projected with velocity $\overrightarrow{\mathrm{v}}= \left(\mathrm{\alpha }\hat{\mathrm{i}} + \mathrm{\beta }\hat{\mathrm{j}}\right)$ m/s. The time of flight of body considering x as horizontal and y as the vertical axis and g is the acceleration due to gravity:

Two balls are projected along the same initial line from a height h = 20 m. The direction of the projections is horizontal. If initial speeds are 20 m/s and 25 m/s, the maximum separation between the balls when they will hit the ground is

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?

If three coordinates of a particle change according to the equations $\mathrm{x}=3{\mathrm{t}}^2, \mathrm{y}=2\mathrm{t}, \mathrm{z}=4$, then the magnitude of the velocity of the particle at time t = 1 second will be

Two bodies of equal masses are projected, one from top of the tower in the horizontal direction and other from the foot of tower at an angle of 45° with the horizontal, then the acceleration of their center of mass is

A projectile is projected at an angle of 60° from horizontal. At an instant, it is moving at an angle of 30° with horizontal with velocity 10 m/s. The radial acceleration of the particle at that instant is:

The initial velocity of a projectile is given by $\mathrm{v} = \left(20\hat{\mathrm{i}} + 50\hat{\mathrm{j}} + 70\hat{\mathrm{k}}\right)$ m/s. If acceleration due to gravity is g = 10($-\hat{\mathrm{k}}$) m/${\mathrm{s}}^2$, then the time after which projectile returns to the same horizontal level is:

Two persons start moving along two crossroads with the same speed 5 m/s. One in the direction of north and other is in the direction of the east as shown in the figure. Angular velocity of A with respect to B is :

In projectile motion, accelerations of the projectile when it is gaining height and losing height respectively are

A person, who can swim with speed u relative to water, wants to cross a river (of width d and water is flowing with speed v). The minimum time in which the person can do so is:

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:

A particle is projected on a horizontal surface with kinetic energy K at an angle $\theta$ with the vertical. The kinetic energy of the particle at the highest point of trajectory is:

Velocity and acceleration vectors of a particle moving on a circular path are shown here. At the instant shown:

A cricketer can throw a ball to a maximum horizontal distance of 50 m. How much high above the ground can he throw the same ball?

Coordinates of a particle as a function of time t are x = 2t, y = 4t. It can be inferred that the path of the particle will be