Crossing the River(Minimum Time)

Q. A river is flowing from west to east at a speed of $5\text{m/min}$. A man on the south bank of the river, capable of swimming at $10\text{m/min}$ in still water, wants to swim across the river in the shortest time. He should swim in a direction

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

Q.

The contrapositive of the statement “If I reach the station in time, then I will catch the train” is

1. If I will catch the train, then I reach the station in time.

2. If I do not reach the station in time, then I will catch the train.

3. If I do not reach the station in time, then I will not catch the train.

4. If I will not catch the train, then I do not reach the station in time.

Q. A man wishes to cross a river in a boat. If he crosses the river in minimum time, he takes $10$ minutes with a drift of $120m$. If he crosses the river taking shortest route, he takes $12.5$ minutes, find the velocity of the boat with respect to water.

1. $20m/min$
2. $12m/min$
3. $10m/min$
4. $8m/min$

Q. A man is capable of swimming at $10\text{m/s}$ in still water. If he has to cross a river flowing at $5\text{m/s}$ and reach the opposite shore in the least possible time, how far along the river will he drift if distance between the two shore is $100\text{m}$.

1. $5\text{m}$
2. $10\text{m}$
3. $50\text{m}$
4. $100\text{m}$

Q. A river flows $3{\text{km h}}^{-1}$ and a man is capable of swimming $2{\text{km h}}^{-1}$. He wishes to cross it such that displacement parallel to river is minimum. In which direction will he swim?

1. ${\sin }^{-1}\left(\frac{2}{3}\right)$
2. ${\cos }^{-1}\left(\frac{2}{3}\right)$
3. ${\cot }^{-1}\left(\frac{2}{3}\right)$
4. ${\tan }^{-1}\left(\frac{2}{3}\right)$

Q. A river is flowing from east to west at a speed of $5m/min$. A man on south bank of river, capable of swimming $10m/min$ in still water wants to swim across the river in shortest time; he should swim

1. due north
2. due north-east
3. due north-east with double the speed of river
4. none of the above

Q. A man can row a boat at $4km/h$ in still water. What is the minimum possible time in which he can cross the river if the speed of the current is $4km/h$ and the width of the river is $8km$?

1. $1h$
2. $2h$
3. $\sqrt{3}h$
4. $\sqrt{2}h$

Q. A man wants to cross the river to an exactly opposite point on the other bank. If he can row his boat with $\frac{2}{\sqrt{3}}$ times 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.

A river is flowing from west to east at a speed of 5 metres per minute. A man on the south bank of the river, capable of swimming at 10 metres per minute in still water, wants to swim across the river in the shortest time. He should swim in a direction

1. Due north

2. ${30}^{\circ }$ east of north

3. ${30}^{\circ }$ west of north

4. ${60}^{\circ }$ east of north

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

1. $5\sqrt{2}h$
2. $10\sqrt{2}h$
3. $0h$
4. $5h$

Q. A river $5\sqrt{3}km$ wide flows with a velocity of $5kmph$. A boat which can move with a velocity of $10kmph$ in still water crosses the river, without any drift. For this, the boat has to tilt its velocity by $\theta$ with the straight path in the right direction (see figiure). If the boat's velocity is tilted by the same angle $\theta$ in the wrong direction, the drift will be

1. $8.1km$
2. $9.1km$
3. $10km$
4. $12km$

Q. A stone is projected from the ground with a velocity of $25m/s$. $2$ seconds later, it just clears wall $5m$ high. Then angle of projection of the stone and the greatest height reached are- $[$Neglect air resistance, Assume $g=10m/se{c}^2$$]$

1. $30$, $7.8m$
2. $30$, $8.7m$
3. $60$, $8.7m$
4. $60$, $8.7cm$

Q. Two stones are projected with the same speed but making different angles with the horizontal. Their horizontal ranges are equal. The angle of projection of one is $\frac{\pi }{3}$ and the maximum height reached by it is $102m$. Then the maximum height reached by the other is:

1. $336m$
2. $224m$
3. $34m$
4. $56m$

Q. A swimmer wishes to cross a $500m$ wide river flowing at $5Km/h$. His speed with respect to water is $3Km/h$
(a) If he heads in a direction making an angle $\theta$ with the flow, find the time he takes to cross the river.

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

Q. A river is flowing from west to east with a speed 5 $m{s}^{-1}$. A swimmer can swim in still water at a speed of 10 $m{s}^{-1}$. If he wants to start from point A on south bank and reach opposite point B on north bank, in what direction should he swim?

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

Q. A boat crossing a river moves with a velocity v relative to still water. The river is flowing with a velocity $v/2$ with respect to the bank. The angle with respect to the flow direction with which the boat should move to minimize the drift is$?$

1. ${30}^o$
2. ${150}^o$
3. ${60}^o$
4. ${120}^o$

Q. A block at rest slides down a smooth inclined plane which makes and angle ${60}^o$ with the vertical and it reaches the ground in $t_1$ seconds. Another block is dropped vertically from the same point and reaches the ground in $t_2$ seconds. What is the ratio of $t_1:t_2$?

Q. A man can row a boat with $4km/h$ in still water. If he is crossing a river where the current is $2km/h$ and width of river is $4km$. How long will it take him to row $2km$ up the stream and then back to his starting point?

1. $2hrs$
   
2. $1hr$
   
3. $\frac{4}{3}hrs$
   
4. None of the above
   

Q. If $R$ and $H$ represent horizontal range and maximum height of the projectile, then the angle of projection with the horizontal is:

1. $ta{n}^{-1}\left(H/R\right)$
2. $ta{n}^{-1}\left(4R/H\right)$
3. $ta{n}^{-1}\left(2H/R\right)$
4. $ta{n}^{-1}\left(4H/R\right)$

Q.

A river is flowing from west to east at a speed of 5 metres per minute. A man on the south bank of the river, capable of swimming at 10 metres per minute in still water, wants to swim across the river in the shortest time. He should swim in a direction

1. Due north

2. ${30}^{\circ }$ east of north

3. ${30}^{\circ }$ west of north

4. ${60}^{\circ }$ east of north

Q.

A river is flowing with a speed of 1 km/hr. A swimmer wants to go to point ‘C’ starting from ‘A’. He swims with a speed of 5 km/hr at an angle $\theta$ w.r.t the river flow. If AB =BC =400 m, what is the value of $\theta$?

1. ${37}^{\circ }$

2. ${53}^{\circ }$

3. ${30}^{\circ }$

4. ${60}^{\circ }$

Q. A river $5\sqrt{3}km$ wide flows with a velocity of $5kmph$. A boat which can move with a velocity of $10kmph$ in still water crosses the river, without any drift. For this, the boat has to tilt its velocity by $\theta$ with the straight path in the right direction (see figiure). If the boat's velocity is tilted by the same angle $\theta$ in the wrong direction, the drift will be

1. $8.1km$
2. $9.1km$
3. $10km$
4. $12km$

Q. A boat crossing a river moves with a velocity $\overrightarrow{v}$ relative to still water. The river is flowing with a velocity $\overrightarrow{v}/2$ with respect to the bank. The angle at which the boat should start from the bank with respect to the river, to minimize the drift is:

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${15}^{\circ }$
4. ${90}^{\circ }$

Q. Suppose Swarglok (heaven) is in constant motion at a speed of 0.9999c with respect to the earth. According to the earth's frame, how much time passes on the earth before one day passes on Swarglok?

Q. A ball is thrown horizontally from a cliff such that it strikes ground after $5$ sec. The line of sight from the point of projection to the point of hitting makes an angle of ${37}^0$ with the horizontal. The initial velocity of the projection is $u$ m/sec. Find value of $u/10$ and round off to the nearest integer. (Take $g=10m/s^2$)

Q. A stone is projected with a velocity $20\sqrt{2}m/s$ at an angle of ${45}^o$ to the horizontal. The average velocity of stone during its motion from starting point to its maximum height is: $(g=10m/s^2)$

1. $10\sqrt{5}m/s$
2. $20\sqrt{5}m/s$
3. $5\sqrt{5}m/s$
4. $20m/s$

Q.

A river is flowing with a speed of 1 km/hr. A swimmer wants to go to point ‘C’ starting from ‘A’. He swims with a speed of 5 km/hr at an angle $\theta$ w.r.t the river flow. If AB =BC =400 m, what is the value of $\theta$?

1. ${37}^{\circ }$

2. ${53}^{\circ }$

3. ${30}^{\circ }$

4. ${60}^{\circ }$

Q. If the boat crosses the river in shortest possible time, then its drift will be:-

1. $250m$
2. zero
3. $125m$
4. $\frac{250}{\sqrt{3}}m$

Q. A swimmer dived off a cliff with a running horizontal leap. What must his minimum speed be just as he leaves the top of the cliff so that he will miss the ledge at the bottom which is $2m$ wide and $9m$ below the top of the cliff? (in m/s)

1. $9$
   
2. $1.4$
   
3. $1$
   
4. $2$