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# River man problem- Different cases, practice problems, FAQs

It is a bright sunny day and Nimit decides to take a swim in the river. He changes into his swimming apparel and goes to the riverbank where he usually goes for swimming. The river is quiet and calm. He jumps into the river and starts from point A, performing strokes perpendicular to the river, swimming in a straight line, reaching the other side of the bank at point B exactly opposite to point A. Now the river starts to flow with a certain speed, now he is no longer able to swim in a straight line with his previous approach. In other words, now his direction of swimming makes an angle with the direction of the flow of the river. After reaching the other bank, he notices that he didn’t reach point B. Why does this happen? What difference did the direction in which he swam make? Let’s understand all such cases in detail!

• River swimmer problem
• Time taken to cross the river downstream
• Time taken to cross the river upstream
• Swimming perpendicular to to the river, drift
• Crossing the river perpendicularly without drift
• Practice problems
• FAQs

## River swimmer problem

In a river swimmer problem, there will be different cases when a swimmer tries to cross the river. When the river does not flow, he is able to swim in a straight line along AB shown in figure. But when the river starts to flow, he is forced to swim along path AC, if he intends to swim along AB. The distance BC is called drift.

Let u- speed of the river.

v- speed of the man in still water.

vrg- velocity of river with respect to ground.

vmr- velocity of man with respect to river= velocity of man in still water.

vmg- velocity of man with respect to ground.

Let vmr, vrg and vmg form three sides of a triangle. vmr and vrg if added, then their vector addition gives the value of vmg. This is called the vector law of addition.

## Time taken to cross the river downstream

Swimming downstream refers to going in the same direction as the river.

Here

The time taken to travel a distance d is given by,

${t}_{AB}=\frac{d}{\left|\stackrel{\to }{{v}_{mg}}\right|}=\frac{d}{v+u}$

## Time taken to cross the river upstream

Swimming perpendicular to the river, drift

Let the swimmer jump into the river at point A and swim perpendicular to the river. The river flows horizontally with a velocity from west to east with a velocity vrg as shown. Due to this, the man will not reach exactly at point B, a “drift” will occur and he would end up at point C.

Let tAC indicate the time taken to cover the distance AC.

## Crossing the river perpendicularly without drift

To reach a point perpendicularly opposite to the river, one must swim at an angle to the river to account for the drift of the river. Let be the angle with the vertical to reach the point B.

vx=u-vsin is the horizontal component of velocity.

The drift in this case,${x}_{drift}={v}_{x}×t=0$

Substituting  we get

For minimum time, the man must swim perpendicular to the river flow.

${t}_{min}=\frac{d}{v}$

## Practice problems

Q. The velocity of the boat is 5 km/hr in still water. If the velocity of the boat w.r.t. ground is 3 km/hr, then the velocity of the river will be

(a)4.5 km/hr (b)4 km/hr (c)1.5 km/hr (d)3 km/hr

A. d

Given, vbr=5 km/hr, vbg=4 km/hr

From fig, it is evident that,

Q. A man wants to cross a river using a boat. If he crosses the river in a minimum time, it takes 10 min with a drift of 120 m. It takes 12.5 min for him to cross the river along the shortest route.

Find the velocity of the boat with respect to water.

Given,

Drift,x =120 m, Time, tmin =10 min

Let,

vbr- velocity of boat wrt river.(Also called velocity of boat in still water)

vrg- Velocity of river wrt ground

| vrg|=u|vbr|=v

The minimum time needed to cross the river tmin=d vbr=d v

d=10 v

Now, x= vrgt 120=10 u

vrg=u=12 m/min

Along the shortest route;

Time=12.5 min

For zero drift, the time taken to cross the river,

t=dv2-u212.5 =10vv2-122

v=20 m/min.

Q.A man can row a boat with 4 km/hr in still water. If the speed of the current is 2 km/hr, in what direction should he swim to reach the opposite point?

A.

Given, vbr=4 km/hr, vrg=2 km/hr

$sin\theta =\frac{{v}_{rg}}{{v}_{br}}=\frac{2}{4}=\frac{1}{2}⇒\theta ={sin}^{-1}\left(\frac{1}{2}\right)={30}^{0}$

He should swim at an angle of 300+900=1200 with the direction of current.

Q. Water is flowing in the west-east direction at a speed of 8 m/min. A man starts from the South bank of the river, swimming at the rate of 20 m/min in still water. He wants to swim across the river in the shortest time. In what direction, should he swim?.

(a)Due north (b)300 East of North (c)300 West of North

(d) 600 East of North

A. a

Time taken to cross the river,

$t=\frac{d}{{v}_{s}cos\theta }$

For time to be minimum, cos should be maximum

$⇒\theta ={0}^{0}$

Therefore, the swimmer should swim North.

## FAQs

Q. What is the difference between speed and velocity?
A.
Speed is a scalar quantity and it carries only magnitude. Velocity on the other hand is a vector which carries both magnitude and direction.

Q. Write the dimensional formula for velocity.
A.
$Velocity=\frac{Displacement}{Time}=\left[L\right]\left[T\right]=\left[LT-1\right]$

Q. Write the expression for speed of a boat in the downstream direction.
A.
Speed downstream= speed of river+ speed of boat in still water.

Q. Write the expression for speed of a boat in the upstream direction.
A.
Speed upstream= speed of boat in still water - speed of the river.

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