Relative Velocity in Two Dimensions

A boat crosses a river from one bank A to another bank B which is opposite. The distance between them is D. The speed of water is $V_W$ and that of boat relative to water is $V_B$. If $V_B = 2V_W$, the time taken by the boat to cross the river directly along AB is $\left(\sin 30^\circ = \frac{1}{2}\right) \cdot \left(\cos 30^\circ = \frac{\sqrt{3}}{2}\right)$

A river of width 200 m is flowing from west to east with a speed of 18 km/h. A boat, moving with speed of 36 km/h in still water, is made to travel one-round trip (bank to bank of the river). Minimum time taken by the boat for this journey and also the displacement along the river bank are ______ and ______ respectively.

Two boys are standing at points A and B on ground where distance AB = a. The boy at point B starts running perpendicular to line AB with velocity '$V_1$ '. The boy at point A starts running simultaneously with velocity 'V' and catches the other boy in time 't'. The value of 't' is

Airplanes A and B are flying with constant velocity in the same vertical plane at angles $30^\circ$ and $60^\circ$ with respect to the horizontal as shown in the figure. The speed of A is $100\sqrt{3}$ m/s. At time $t = 0$ s, an observer in A finds B at a distance of $500$ m. This observer sees B moving with a constant velocity perpendicular to the line of motion of A. If at $t = t_0$, A just escapes being hit by B, $t_0$ in second is

$\{\sin(30^\circ) = 0.5, \cos(30^\circ) = 3^{0.5}/2\}$

A river is flowing from west to east direction with speed of $9\text{ km h}^{-1}$. If a boat capable of moving at a maximum speed of $27\text{ km h}^{-1}$ in still water, crosses the river in half a minute, while moving with maximum speed at an angle of $150^\circ$ to direction of river flow, then the width of the river is:

The maximum speed of a boat in still water is 27 km/h. Now this boat is moving downstream in a river flowing at 9 km/h. A man in the boat throws a ball vertically upwards with speed of 10 m/s. Range of the ball as observed by an observer at rest on the bank is __________ cm. (Take $g = 10$ m/s2)

Two girls are standing at the ends ' $A$ ' and ' $B$ ' of a ground where $AB = b$. The girl at ' B ' starts running in a direction perpendicular to ' $AB$ ' with velocity ' $V_1$ '. The girl at ' A ' starts running simultaneously with velocity ' $V_2$ ' and in shortest distance meets the other girl in time ' $t$ '. The value of ' $t$ ' is

Two boys are standing at points A and B on ground, where distance $AB = x$. The boy at B stars running perpendicular to $AB$ with velocity $v_1$. The boy at A starts running simultaneously with velocity v and meets the other boy in time $t$. The value of $t$ is

The speed of a swimmer is $4\text{ km h}^{-1}$ in still water. If the swimmer makes his strokes normal to the flow of river of width $1\text{ km}$, he reaches a point $750\text{ m}$ down the stream on the opposite bank. The speed of the river water is ___________ $\text{km h}^{-1}$

A girl standing on road holds her umbrella at 45$^\circ$ with the vertical to keep the rain away. If she starts running without umbrella with a speed of 15$\sqrt{2}$ kmh${}^{-1}$, the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is:

A fighter jet is flying horizontally at a certain altitude with a speed of 200 ms${}^{-1}$. When it passes directly overhead an anti-aircraft gun, a bullet is fired from the gun, at an angle $\theta$ with the horizontal, to hit the jet. If the bullet speed is 400 m/s, the value of $\theta$ will be ___________$^\circ$.

A swimmer can swim with velocity of 12 km/h in still water. Water flowing in a river has velocity 6 km/h. The direction with respect to the direction of flow of river water he should swim in order to reach the point on the other bank just opposite to his starting point is ____________$^\circ$. (Round off to the Nearest Integer) (Find the angle in degrees)

A bomb is dropped by fighter plane flying horizontally. To an observer sitting in the plane, the trajectory of the bomb is a:

A person is swimming with a speed of 10 m/s at an angle of 120$^\circ$ with the flow and reaches to a point directly opposite on the other side of the river. The speed of the flow is 'x' m/s. The value of 'x' to the nearest integer is __________.

A butterfly is flying with a velocity $4\sqrt{2}$ m/s in North-East direction. Wind is slowly blowing at 1 m/s from North to South. The resultant displacement of the butterfly in 3 seconds is:

A swimmer wants to cross a river from point A to point B. Line AB makes an angle of 30$^\circ$ with the flow of river. Magnitude of velocity of the swimmer is same as that of the river. The angle $\theta$ with the line AB should be _________$^\circ$, so that the swimmer reaches point B.

When a car is at rest, its driver sees rain drops falling on it vertically. When driving the car with speed v, he sees that rain drops are coming at an angle 60° from the horizontal. On further increasing the speed of the car to (1 + $\beta$)v, this angle changes to 45o. The value of $\beta$ is close to:

The speed of a swimmer in still water is 20 m/s. The speed of river water is 10 m/s and is flowing due east. If he is standing on the south bank and wishes to cross the river along the shortest path, the angle at which he should make his strokes w.r.t. north is given by:

The stream of a river is flowing with a speed of 2km/h. A swimmer can swim at a speed of 4km/h. What should be the direction of the swimmer with respect to the flow of the river to cross the river straight?

A person standing on an open ground hears the sound of a jet aeroplane, coming from north at an angle 60o with ground level. But he finds the aeroplane right vertically above his position. If v is the speed of sound, speed of the plane is:

Ship A is sailing towards north-east with velocity$\vec{v} = 30\hat{i} + 50\hat{j}$ km/hr where $\hat{i}$ points east and $\hat{j}$, north. Ship B is at a distance of 80 km east and 150 km north of Ship A and is sailing towards west at 10 km/hr. A will be at minimum distance from B in:

A man in a car at location Q on a straight highway is moving with speed $v$. He decides to reach a point P in a field at a distance d from the highway (point M) as shown in the figure. Speed of the car in the field is half to that on the highway. What should be the distance RM, so that the time taken to reach P is minimum?

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

The maximum range of a bullet fired from a toy pistol mounted on a car at rest is $R_0 = 40$ m. What will be the acute angle of inclination of the pistol for maximum range when the car is moving in the direction of firing with uniform velocity $v = 20$ m/s on a horizontal surface? ($g = 10$ m/s$^{2}$)

Two boys are standing at the ends A and B of a ground where AB = a. The boy at B starts running in a direction perpendicular to AB with velocity v1. The boy at A starts running simultaneously with velocity v and catches the other in a time t, where t is

A ball is thrown from a point with a speed ν0 at an angle of projection θ. From the same point and at the same instant person starts running with a constant speed $\frac{v_0}{2}$ to catch the ball. Will the person be able to catch the ball? If yes, what should be the angle of projection θ?

The width of river is 1 km. The velocity of boat is 5 km/hr. The boat covered the width of river along the shortest path in 15 min. Then the velocity of river stream is

A person swims in a river aiming to reach exactly opposite point on the bank of a river. His speed of swimming is $0.5\text{ m/s}$ at an angle $120^\circ$ with the direction of flow of water. The speed of water in stream is

The speed of a boat is $5\text{ km/h}$ in still water. It crosses a river of width $1.0\text{ km}$ along the shortest possible path in $15\text{ min}$. The velocity of the river water is (in $\text{km/h}$)

A boat is sent across a river with a velocity of $8\text{ km h}^{-1}$. If the resultant velocity of boat is $10\text{ km h}^{-1}$, then velocity of river is

A boat is sent across a river with a velocity of $8\text{ km h}^{-1}$. If the resultant velocity of boat is $10\text{ km h}^{-1}$, then velocity of river is