Relative Velocity in 2D

Q.

A boat takes $2hr$ to travel $8km$ and back in still water. If the velocity of water is $4km/h$r, the time taken for going upstream $8km$ and coming back is:

1. $2hr$

2. $2hr40min$

3. $1hr20min$

4. Cannot be estimated with the information given

Q. Two cars $A$ and $B$ are $5\text{km}$ apart on a line joining South to North. Car $A$ further North is streaming west at $10\text{km/hr}$ and Car $B$ is streaming North at $10\text{km/hr}$. What is their closest distance of approach?

1. $5\text{km}$
2. $\frac{5}{\sqrt{2}}\text{km}$
3. $2\text{km}$
4. $\sqrt{3}\text{km}$

Q. A man can row a boat at $4\text{km/h}$ in still water. If he is crossing a river which is $4\text{km}$ wide, and the speed of current is $2\text{km/h}$, then in what direction should his boat be headed if he wants to reach a point on the other bank, directly opposite to the starting point?

1. ${30}^{\circ }$ with direction of river flow
2. ${60}^{\circ }$ with direction of river flow
3. ${120}^{\circ }$ with direction of river flow
4. ${150}^{\circ }$ with direction of river flow

Q. Suppose two particle, $1$ and $2$ are projected in vertical plane simultaneously with the velocities of $160\text{m/s}$ and $100\text{m/s}$ respectively as shown in the figure.


Their angles of projection are ${30}^{\circ }$ and $\theta$, respectively, with the horizontal. If they collide after time $t$ in air, then which of the following is incorrect?

1. $\theta ={53}^{\circ }$ and they will have same speeds just before the collision.
2. $\theta ={53}^{\circ }$ and they will have different speeds just before the collision.
3. $x<(1280\sqrt{3}-960)\text{m}$
4. It is possible that the particles collide when both of them are at their highest point.

Q.

Three particles A, B and C are situated at the vertices of an equilateral triangle ABC of side 'd' at t = 0. Each of the particles moves with constant speed v. A always has its velocity along AB, B along BC and C along CA. At what time will the particles meet each other?

1. $\frac{2d}{3v}$

2. $\frac{2d}{\sqrt{3}v}$

3. $\frac{d}{\sqrt{3}v}$

4. $insufficientdata$

Q. The stream of a river is flowing at a speed of $2km/h$. A swimmer can swim at a speed of $4km/h$ in still water. What should be the direction of the swimmer with respect to the flow of the river to cross the river with minimum drift?

1. ${60}^{\circ }$
2. ${120}^{\circ }$
3. ${90}^{\circ }$
4. ${150}^{\circ }$

Q. A swimmer wishes to cross a river that is $500m$ wide flowing at a rate $u$. For this, he makes an angle $\theta$ with the vertical to the direction of flow of the river as shown. His speed with respect to still water is $v$. In order to cross the river in minimum time, the value of $\theta$ in degrees should be

1. $0^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${90}^{\circ }$

Q. A gun moving at a speed of $30m/s$ fires a bullet (in the same direction gun is moving) at an angle ${30}^o$ with the horizontal and a velocity of $150m/s$ relative to the gun. Find the distance between the gun and the bullet when the bullet hits the ground. (Take $g=10m/s^2$)

1. $1500m$
2. $2400m$
3. $1950m$
4. $1000m$

Q. An aeroplane has to go from a point $A$ to $B$ due north. Wind is blowing due east at a speed of $250\text{km/h}$. If the air speed of plane is $500\text{km/h}$ then its direction of head to reach point $B$ is :

1. ${15}^{\circ }$ west of north
2. ${60}^{\circ }$ west of north
3. ${45}^{\circ }$ west of north
4. ${30}^{\circ }$ west of north

Q. An airplane has to go from a point $A$ to another point $B$, $500\text{km}$ away due ${30}^{\circ }$ East of North. Wind is blowing due North at speed of $20\text{m/s}$. The speed of the airplane is $150\text{m/s}$. Find the direction in which the pilot should head the plane so as to reach the point $B$. Also find the time taken by the plane to go from $A$ to $B$.

1. $\left({\sin }^{-1}\frac{1}{15}\right)$ East of North, $50\text{min}$
2. $({30}^{\circ }+{\sin }^{-1}\frac{1}{15})$ East of North, $50\text{min}$
3. $({60}^{\circ }+{\sin }^{-1}\frac{1}{15})$ East of North, $50\text{min}$
4. $({30}^{\circ }+{\sin }^{-1}\frac{1}{15})$ East of North, $60\text{min}$

Q. A swimmer wishes to cross a $1\text{km}$ wide river flowing at a speed of $5{\text{kmh}}^{-1}$. His speed in still water is $3{\text{kmh}}^{-1}$. He has to reach the directly opposite point in minimum possible time. If he does not reach the directly opposite point by swimming, he has to walk the extra distance at $5{\text{kmh}}^{-1}$. Find the minimum time taken by the swimmer to reach the desired point. Use the following data if needed:
$\sqrt{91}=9.5,\frac{1}{2.85}\simeq 0.35$

1. $0.64\text{hr}$
2. $1\text{hr}$
3. $0.33\text{hr}$
4. Not possible

Q. An aircraft flies at a speed of $800\text{km/h}$ in still air. The wind is blowing at $400\sqrt{2}\text{km/h}$ from south direction and the pilot wishes to travel from point $A$ to point $B$, North-East of $A$. Find the direction with reference to the North, the pilot must steer and time ($\text{hr}$) of his journey if $AB=1200\text{km}$. (Take $\cos {15}^{\circ }=0.96$)

1. ${30}^{\circ },2.0\text{hr}$
2. ${75}^{\circ },\frac{25}{32}\sqrt{2}\text{hr}$
   
3. ${45}^{\circ },\frac{25}{27}\sqrt{2}\text{hr}$
   
4. ${45}^{\circ },\frac{15}{32}\sqrt{3}\text{hr}$
   

Q. A bird is flying towards north with a velocity 40 km/h and a train is moving with a velocity 40 km/h towards east. What is the velocity of the bird noted by a man in the train?

1. $40\sqrt{2}\text{km/h}N-E$
2. $40\sqrt{2}\text{km/h}S-E$
3. $40\sqrt{2}\text{km/h}N-W$
4. $40\sqrt{2}\text{km/h}S-W$

Q.

A bird is flying due east with a velocity of 4 m/s. The wind starts to blow with a velocity of 3 m/s due north. What is the magnitude of relative velocity of bird w.r.t. wind? Find out the angle it makes with the x-axis?

1. 7m/s, $ta{n}^{-1}\left(\frac{3}{4}\right)$

2. 7 m/s, $ta{n}^{-1}\left(\frac{4}{3}\right)$

3. 5 m/s, $ta{n}^{-1}\left(\frac{3}{4}\right)$

4. 5 m/s, $ta{n}^{-1}\left(\frac{4}{3}\right)$

Q.

Three particles A, B and C are situated at the vertices of an equilateral triangle ABC of side 'd' at t = 0. Each of the particles moves with constant speed v. A always has its velocity along AB, B along BC and C along CA. At what time will the particles meet each other?

1. $\frac{2d}{3v}$

2. $\frac{2d}{\sqrt{3}v}$

3. $\frac{d}{\sqrt{3}v}$

4. $insufficientdata$

Q. Car A has an acceleration of $2m/s^2$ due east and car B, $4m/s^2$ due North. Acceleration of car B with respect to car A is

1. $2\sqrt{5}$ at an angle ${\tan }^{-1}\left(2\right)$ North of West.
2. $2\sqrt{5}$ at an angle ${\tan }^{-1}\left(2\right)$ West of North.
3. $4\sqrt{2}$ at an angle ${\tan }^{-1}\left(\frac{1}{3}\right)$ North of West.
4. $4\sqrt{2}$ at an angle ${\tan }^{-1}\left(\frac{1}{3}\right)$ West of North.

Q. A man can swim with a velocity $v$ relative to water. He has to cross a river of width $d$ flowing with a velocity $u(u>v)$. The distance through which he is carried downstream by the river is $x$. Which of the following statements are correct?

1. If he crosses the river in minimum time, $x=\frac{du}{v}$
2. $x$ cannot be less than $\frac{du}{v}$
3. For $x$ to be minimum, he has to swim in a direction making an angle of $\frac{\pi }{2}+{\sin }^{–1}\left(\frac{v}{u}\right)$ with the direction of the flow of water.
4. $x$ will be maximum if he swim in a direction making an angle of $\frac{\pi }{2}-{\sin }^{–1}\left(\frac{v}{u}\right)$ with the direction of the flow of water.

Q. A man wants to reach point B on the opposite bank of a river flowing at a speed $u$ as shown. What minimum velocity relative to water should the man have so that he can reach directly to point B?

1. $\frac{u}{\sqrt{2}}$, in the upstream at an angle ${45}^{\circ }$ with the vertical
2. $\sqrt{2}u$, in the upstream at an angle ${45}^{\circ }$ with the vertical
3. $\frac{u}{\sqrt{2}}$, in the downstream at an angle ${45}^{\circ }$ with the vertical
4. $\sqrt{2}u$, in the downstream at an angle ${45}^{\circ }$ with the vertical

Q. A man swims from a point A on one bank of a river of width $100m$ . When he swims perpendicular to the water current, he reaches the other bank $50m$ downstream. The angle to the bank at which he should swim, to reach the directly opposite point B on the other bank is

1. ${80}^{\circ }$ with the direction of flow of river
2. ${110}^{\circ }$ with the direction of flow of river
3. ${120}^{\circ }$ with the direction of flow of river
4. ${150}^{\circ }$ with the direction of flow of river

Q.

A boy throws a ball upwards with velocity $v_0$ = 20 m/s . The wind imparts a horizontal acceleration of 4 m/$s^2$ to the left.

The angle $\theta$ at which the ball must be thrown so that the ball returns to the boy's hand is ( g = 10m/$s^2$ )

1. $ta{n}^{-1}\left(\frac{1}{2}\right)$

2. $ta{n}^{-1}\left(0.2\right)$

3. $ta{n}^{-1}\left(2\right)$

4. $ta{n}^{-1}\left(0.4\right)$

Q. A car is moving towards east with a speed of $25{\text{km h}}^{-1}$. To the driver of the car, a bus appears to move towards north with a speed of $25\sqrt{3}{\text{km h}}^{-1}$. What is the actual velocity of bus?

1. $50{\text{km h}}^{-1},{30}^{\circ }$ East of North
2. $50{\text{km h}}^{-1},{30}^{\circ }$ North of East
3. $25{\text{km h}}^{-1},{30}^{\circ }$ East of North
4. $25{\text{km h}}^{-1},{30}^{\circ }$ North of East

Q. A man swims with a speed of $4km/h$ in still water. The speed of the river is $3km/h$. He starts swimming in the river perpendicular to the direction of the flow of the current of the river. The river is $2km$ wide. What is the horizontal shift of the man?

1. $0.75km$
2. $1.5km$
3. $2km$
4. $1km$

Q.

A river flows due south with a speed of 2.0 m/s. A man steers a motorboat across the river; his velocity relative to the water is 4 m/s due east. The river is 800 m wide. How much time is required to cross the river?

1. 187.6 s

2. 400 s

3. 173.91 s

4. 200 s

Q. A man wants to cross the river to an exactly opposite point on the other bank. If he can row his boat with twice the velocity of the current, then at what angle to the current he must keep the boat pointed?

1. ${60}^{\circ }$
2. ${90}^{\circ }$
3. ${120}^{\circ }$
4. ${150}^{\circ }$

Q. If ${\overrightarrow{v}}_{m\omega }$ = velocity of a man relative to water $\overrightarrow{v_{\omega }}=$ velocity of water, $\overrightarrow{v_m}$ = velocity of man relative to ground, match the following.
$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{i. Minimum distance for}{v}_{m\omega }>v_{\omega }& a.\theta =si{n}^{-1}\left(\frac{v_{m\omega }}{v_{\omega }}\right)\\ \text{ii Minimum time for}{v}_{m\omega }\ge v_{\omega }& b.\overrightarrow{v_m}\perp \overrightarrow{v_{\omega }}\\ \text{iii. Minimum distance for}{v}_{m\omega }<v_{\omega }& c.\overrightarrow{v_{m\omega }}\perp \overrightarrow{v_{\omega }}\\ \text{iv. Minimum time for}{v}_{m\omega }<v_{\omega }& d.\theta =si{n}^{-1}\frac{v_{\omega }}{v_{m\omega }}\end{array}$

1. i- b, d ii- c iii- a iv- c
2. i- b, d ii- a iii- a iv- c
3. i- b, d ii- c iii- a iv- a
4. i- b, c ii- c iii- a iv- c

Q. A river is flowing with a speed of $1{\text{kmh}}^{-1}$. A swimmer wants to go to point $\text{C}$ starting form $\text{A}$. He swims with a speed of $5{\text{kmh}}^{-1}$ at an angle $\theta$ with the river flow. If $\text{AB}=\text{BC}=400\text{m}$, at what angle with the river bank should the swimmer swim?

1. $\theta =\frac{1}{2}{\sin }^{-1}\frac{24}{25}$
2. $\theta =\frac{1}{2}{\sin }^{-1}\frac{12}{25}$
3. $\theta =\frac{1}{2}{\sin }^{-1}\frac{16}{25}$
4. $\theta =\frac{1}{2}{\sin }^{-1}\frac{9}{25}$

Q. A car is moving towards east with a speed of $25{\text{km h}}^{-1}$. To the driver of the car, a bus appears to move towards north with a speed of $25\sqrt{3}{\text{km h}}^{-1}$. What is the actual velocity of bus?

1. $50{\text{km h}}^{-1},{30}^{\circ }$ East of North
2. $50{\text{km h}}^{-1},{30}^{\circ }$ North of East
3. $25{\text{km h}}^{-1},{30}^{\circ }$ East of North
4. $25{\text{km h}}^{-1},{30}^{\circ }$ North of East

Q. Two particles $A$ and $B$ are projected simultaneously from the top of two towers of height $10\text{m}$ and $20\text{m}$ respectively. Particle $A$ is projected upwards at an angle of ${45}^{\circ }$ with horizontal at a speed of $10\sqrt{2}\text{m/s}$ and particle $B$ is projected horizontally at a speed of $10\text{m/s}$ towards $A$. What is the distance between towers if particles collide in the air ?Take $g=10{\text{m/s}}^2$.

1. $10.5\text{m}$
2. $25\text{m}$
3. $20\text{m}$
4. $35\text{m}$

Q. Two particles $A$ and $B$ are projected simultaneously from the towers of same height as shown in the figure with velocities $V_A=20\text{m/s}$ and $V_B=10\text{m/s}$ respectively. They collide in air after 0.5 seconds. Find '$x$'.

1. $5\text{m}$
2. $\frac{5}{\sqrt{3}}\text{m}$
3. $5\sqrt{3}\text{m}$
4. $10\sqrt{3}\text{m}$

Q. A man can row a boat $4km/hr$ in still water. He is crossing a river where the current is $2km/hr$. If the width of the river is $4km$ , the time taken by him to reach a point directly opposite to him is

1. $\frac{5\sqrt{7}}{4}hr$
2. $\frac{4\sqrt{3}}{5}hr$
3. $\frac{2}{\sqrt{3}}hr$
4. $\frac{3\sqrt{2}}{4}hr$