relative velocity and riverboat problems worksheet

Description This is a simulation of a boat crossing a river. Adjust the direction the boat is aimed, the boat's velocity relative to the river, and the river's velocity relative to the earth. Press the "Run" button to watch the boat's trip across the river. Questions to answer: 1) What direction should the boat be aimed to get to the other side of the river in the least amount of time? 2) What direction does the boat need to be aimed to get to the point directly across the river? 3) If the boat is aimed directly across the river, does the speed of the river's current affect the amount of time it takes the boat to cross the river?

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River Boat Problem in 2D From Relative Velocity - JEE Important Topic

• River Boat Problem Relative Velocity In 2d

A Brief Introduction to Relative Velocity

We come into situations when one or more objects move in a non-stationary frame with respect to another observer. For example: a boat crossing a fast-flowing river or an aeroplane flying in the air encountering wind. In all these cases, we must consider the medium's effect on the item to characterise the object's whole motion. We calculate the relative velocity of the object while doing so, taking into account the particle's velocity as well as the velocity of the medium. Since velocity is a vector, the calculations of relative velocity include vector algebra. We’ll look at the problem of relative velocity in more detail throughout this article and discuss river velocity, river boat problems and solutions to relative velocity.

Examples and Mathematical Formulation of Relative Velocity

We all have encountered relative velocity at some point. Some classic examples of relative velocity include situations where, while travelling by automobile, bus or train, you may see that the trees, buildings and other objects outside are moving backwards. Is it true, however, that they are going backwards? No! You're completely aware that your vehicle is moving while the trees remain motionless on the ground. But, if that's the case, why are the trees travelling backwards? Also, even though they are moving, your fellow passengers appear to be motionless to you. 

This is where the concept of relative velocity comes into play. The passengers appear motionless to you because they are at rest, relative to you. However, to someone on the ground who isn’t moving, your fellow passengers are in motion, relative to them. You are also in motion relative to someone on the ground. Although the person on the ground might not be moving, according to you they are moving backwards and they have a velocity relative to you. 

Let’s consider two objects and name them objects A and B. Suppose object A has velocity v A and object B has velocity v B and they are moving relative to some common stationary frame of reference. This frame of reference could be anything; the ground, a lamppost, a bridge, etc. 

Relative velocity is just the difference between the velocities of the objects. Already emphasised before, this difference is not the ordinary difference because velocities are vectors. So, they will follow the rules of vector algebra. 

The velocity of object A relative to B is represented as v AB . The formula for this is:

$v_{A B}=v_{A}-v_{B}$

Similarly, the velocity of object B relative to A is represented by v BA  and its formula is:

$v_{B A}=v_{B}-v_{A}$

From the expressions of v AB and v BA , we can say that they both are additive inverses of each other. This means that:

$v_{A B}=-v_{B A}$

This means that v AB has a direction that is opposite to v BA . Even though they have opposite directions, their magnitude remains the same. 

$\left|v_{A B}\right|=\left|v_{B A}\right|$

River Boat Problem in 2D

When a boat is moving through a river, it is affected by the velocity of the water. The directions of the velocities of the boat and the river are usually different. The motion of the boat is influenced by the relative velocity between them. As usual, the concept of relative velocity will be applied and then the problems will be solved accordingly. 

If a motorboat was heading straight across a river, it would not reach the point exactly opposite to where it started from. This is due to the river current that influences its motion.

Let’s suppose that we have a motorboat which is moving with a velocity of 6 $\dfrac{m}{s}$ directly across the river. If the river has a velocity downstream, the actual resultant velocity of the motorboat will not be the same as it was initially. The velocity of the boat will be a bit more than 6 $\dfrac{m}{s}$ and it will not be in the direction straight across the river, but at some direction downstream with a certain angle. 

The shortest path in the river boat problems is when the boat moves perpendicular to the river current. This whole situation will become clear with some numerical examples that we’ll see in the next section. 

To solve any river boat problem, two things are to be kept in mind.

A boat's speed with respect to the water is the same as its speed in still water.

The velocity of the boat relative to water is equal to the difference in the velocities of the boat relative to the ground and the velocity of the water with respect to the ground. If v BW is the velocity of the boat with respect to the water, and v B , v W are the velocities of the boat and water with respect to the ground respectively, then:

$v_{B W}=v_{B}-v_{W}$

Boat and River Current Velocities

Crossing The River in The Shortest Time

Schematic Diagram of a Boat Going Across a River

Suppose that u is the velocity of the river and v is the velocity of the boat. The boat moves at some angle $\theta$ with respect to the horizontal as shown in the figure. The total velocity of the boat will be the sum of the velocity of the boat with respect to the ground and the velocity of the river. This will be given as

$\begin{align} &v_{b}=\vec{v}+\vec{u} \\ \\ &v_{b}=-v \cos \theta \hat{i}+v \sin \theta \hat{j}+u \hat{i} \\ \\ &v_{b}=(-v \cos \theta+u) \hat{i}+v \sin \theta \hat{j} \end{align}$

The boat needs to move in the vertical direction in order to make it across the river so only the vertical component of the velocity will be used in getting it across the river. The vertical component is $v\sin{\theta}$. The width of the river is d and so the time taken to cross the river will be

$t=\dfrac{d}{v\sin{\theta}}$

For a minimum time 

$\sin{\theta}=1$ 

$\theta=90^{\circ}$

This means that the minimum time to cross the river will be

$t_{min}=\dfrac{d}{v}$

Crossing the River Along the Shortest Path

Schematic Diagram of A Boat Going Across The River With Some Drift

For the boat to go across the river along the shortest path, the drift x should be minimum or more precisely zero. The drift x will be zero when the velocity in the i direction will be zero. This means that

$\begin{align} &u-v \cos \theta=0 \\ \\ &v \cos \theta=u \\ \\ &\cos \theta=\dfrac{u}{v} \\ \\ &\theta=\cos ^{-1}\left(\dfrac{u}{v}\right) \end{align}$

So in order for the boat to go along the shortest path it has to go at an angle of $\theta=\cos^{-1}\left(\dfrac{u}{v}\right)$ with the vertical. 

The time for the shortest path will be given as

Now we have $\cos{\theta}=\dfrac{u}{v}$ and we know that

$\begin{align} &\sin ^{2} \theta+\cos ^{2} \theta=1 \\ \\ &\sin ^{2} \theta=1-\cos ^{2} \theta \\ \\ &\sin \theta=\sqrt{1-\cos ^{2} \theta} \end{align}$

Putting the value of $\sin{\theta}$ will give

$\begin{align} &\sin \theta=\sqrt{1-\left(\dfrac{u}{v}\right)^{2}} \\ \\ &\sin \theta=\sqrt{1-\dfrac{u^{2}}{v^{2}}} \\ \\ &\sin \theta=\sqrt{\dfrac{v^{2}-u^{2}}{v^{2}}} \\ \\ &\sin \theta=\dfrac{\sqrt{v^{2}-u^{2}}}{v} \end{align}$

Inserting this in the expression for time gives

$\begin{align} &t=\dfrac{d}{v\left(\dfrac{\sqrt{v^{2}-u^{2}}}{v}\right)} \\ &t=\dfrac{ d}{\sqrt{v^{2}-u^{2}}} \end{align}$

This is the time taken along the shortest path.

Numerical Examples of Relative Velocity River Boat Problems

Example 1: A boat has a velocity of 10$\dfrac{km}{hr}$ in still water and it crosses a river of width 2 km. If the boat crosses the river along the shortest path possible in 30 minutes, calculate the velocity of the river water.

Solution: 

Since it is given that the boat crosses the river in the shortest path possible, it means that the boat moves perpendicular to the river current. 

Now, we have:

v BW = 10 $\dfrac{km}{hr}$, v w =? .

The distance d=2 km.

Time taken is, t=20 min= 0.5 hr.

We know that the velocity of the boat in respect to water is:

Since the boat is moving perpendicular to the water, we can apply Pythagoras theorem to find the magnitude of the resultant velocity of the boat. 

This means:

$\left|v_{B W}\right|^{2}=\left|v_{B}\right|^{2}+\left|v_{W}\right|^{2}$....(1)

Velocities of Boat and the Water

We have been given that the boat covers a distance of 2km in 0.5 hr. 

This means that the velocity of the boat with respect to the ground will be:

$\begin{align} &v_{B}=\dfrac{2}{0.5} \\ &v_{B}=4 \dfrac{\mathrm{~km}}{ \mathrm{hr}} \end{align}$

Substituting the values in equation (1) we get,

$\begin{align} &10^{2}=4^{2}+\left|v_{W}\right|^{2} \\ \\ &100-16=\left|v_{W}\right|^{2} \\ \\ &\sqrt{84}=v_{W} \\ \\ &9.16 \simeq v_{W} \end{align}$

So, the velocity of the river water is approximately 9.16 $\dfrac{km}{hr}$.

Example 2: The velocity of a boat in still water is 15 $\dfrac{km}{hr}$ and the velocity of the river stream is 10 $\dfrac{km}{hr}$. Find the time taken by the boat to travel 60 km downstream.

S olution: 

Since the boat is travelling downstream, this means that the velocity of the boat and the river have the same direction. 

We have been given:

v BW = 15$\dfrac{km}{hr}$, and v W = 10$\dfrac{km}{hr}$.

Using $v_{B W}=v_{B}-v_{W}$ we can find the value of v B .

v B = v BW + v W

v B = 15 + 10

v B = 25 \[ \dfrac{\mathrm{~km}}{ \mathrm{hr}} \]

The time taken by the boat to travel 60km will then be:

$\begin{align} &t=\dfrac{60}{v_{B}} \\ &t=\dfrac{60}{25} \\ &t=2.4 \mathrm{hr} \end{align}$

The velocity of an object in respect to another object is its relative velocity. It enables us to comprehend how objects move and interact with one another, relative velocity is crucial to understanding Physics. The velocity of an object with respect to another object is its relative velocity. It is a way to gauge how quickly two items are moving in relation to one another. 

For a boat moving along a river or trying to cross a river, the concept of relative velocity is applied. Here, the velocity of the boat and the velocity of the water flow in the river flow are used to calculate the relative velocities. When a boat is moving across a river it moves at some particular angle with respect to the horizontal and evaluating these conditions can tell us the minimum time and the shortest path for the boat to cross the river. 

FAQs on River Boat Problem in 2D From Relative Velocity - JEE Important Topic

1. What is the importance of the river boat problem in JEE Main?

River boat problem is a part of relative velocity. It is confusing at first, but is indeed an important topic for JEE Main . River boat problem is similar to other problems like rain man problems or the aeroplane problems. These problems are also solved using the techniques used in river boat problems. Every year at least 1 question is asked from the kinematics part and the probability of relative velocity being asked is quite high due to the variety of questions that can be framed. 

2. What are absolute and relative velocities?

Relative velocity is the velocity calculated between objects in motion. It depends on the frame of reference of the objects and the observer. It changes with the choice of frame of reference. Absolute velocity on the other hand is the velocity that can be defined with respect to some absolute spatial coordinate system. This velocity will be independent of the frame of reference. We can only measure the relative velocity of any object with our present technology and knowledge about things. 

Relative Motion Worksheet (Motion in Two Dimensions)

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Description

This worksheet is designed for students just beginning to solve "river boat problems". Students use V=d/t to solve for motion in two dimensions. It includes careful scaffolding to help train students to realize two dimensional motion problems are two problems that can be solved independently of one another, with time being "the big link" between the two.

Target Audience: high school Physics

Please look at the preview to see if this is appropriate for your students. The entire product is shown.

To see all my HS level Physics resources, click Here

To see all my Physical Science (IPC) level resources, click Here

What this includes:

• 4 student pages
• Scaffolding to walk students through the step-wise process

If you have any questions, feel free to contact me at [email protected]

The Details:

This is a license for single classroom use. Copyright 2019 Laura Delzer aka "Delzer's Dynamite Designs". Please see preview for more details. This product is covered by federal law. THANK YOU for respecting my hard work!

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Delzer's dynamite designs.

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The Physics Classroom has been devoted to helping students, teachers, and classrooms since the 1990s. We are as passionate about that mission now as we have ever been. If you are a teacher of Physics or Physical Science, we encourage you to use our Video Tutorial with your students. And we also encourage you to consider the use of other resources on our website that coordinate with the video. We have listed a few below to help you get started.  

Teacher Toolkits: Vectors

Curriculum Corner: Vectors and Projectiles

Calculator Pad, Vectors and Projectiles Chapter, Problems #19 - #21

Physics Interactives - River Boat Simulator

Concept Builder: Relative Velocity

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Physics Classroom Tutorial, Vectors and Motion in Two Dimensions Chapter, Relative Velocity and River Boat Problems

Relative Velocity In Two Dimensions, River-boat Problems, Important Topics For JEE 2025

Relative Velocity In Two Dimensions : Learn about relative velocity in two dimensions, a crucial concept in physics that describes the motion of objects relative to each other. Understand vector addition, reference frames, and applications in solving real-world problems.

June 11, 2024

Table of Contents

Relative Velocity in Two Dimensions :  In this section, we describe how observations made by different observers in different frames of reference are related to each other. We find that observers in different frames of reference may measure different positions, velocities, and accelerations for a given particle. That is, two observers moving relative to each other generally do not agree on the outcome of a measurement.

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 Suppose a person riding on a skateboard-throws a ball in such a way that it appears in this person’s frame of reference to move first straight upward and then straight downward along the same vertical line. An observer B on the ground sees the path of the ball as a parabola. Relative to observer B, the ball has a vertical component of velocity (resulting from the initial upward velocity and the downward acceleration due to gravity) and a horizontal component. We will see this relative motion in River-boat problem, aeroplane- wind problems and river-man problems in this section.

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River-boat Problems

While solving problems related to river boat or river swimmer we come across the following terms 

(c)      Velocity of swimmer/boatman with respect to the river v sr or v br

Since,                 v sr = v s – v r

Observing the situation shown in the diagram below, we arrive at some standard results and their special cases.

Since from (1), we get

It  t  is the time taken by the swimmer to cross the river, then

Further if  x  is the drift (displacement along  x  axis) when he reached the opposite bank, then

x  = ( v b ) x t

The relative velocity of swimmer with respect to river must be perpendicular to the river current.

As a result of this the swimmer will be drifted to the point  B .

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Relative Velocity : Condition For Zero Drift  Or Condition To Reach The Opposite Point

Since we calculated the drift value to be x  given by 

     For the above condition to be met, we must have

     A diagrammatic representation is given in support of the above mathematical argument.

Further since 

Also we observe that if  v r  >  v sr  then the swimmer would never reach the point  Q .

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Relative Velocity :  Condition for Minimum Drift

For minimising the drift, we must have

     If we are asked to calculated the angle with the horizontal, then we get

     Also, we conclude that the drift  x  can be minimised only if  v sr  <  v r .

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Aeroplane-wind Problems

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Let us understand with the help of an Example

Ex.      A plane moves in windy weather due east while the pilot points the plane somewhat south of east. The wind is blowing at 50 km/hr directed 30° east of north, while the plane moves at 200 km/hr relative to the wind. What is the velocity of the plane relative to the ground and what is the direction in which the pilot points the plane?

Here we have two unknown quantities (V PG  and θ), so we should not relate the magnitudes of these vectors using triangle law. In such situations we resolve the vectors into components on the coordinate system and then solve the equations for both axes ( x  and  y ).

For the  y  components

v PG ,  y  =  v PW, y  +  v WG, y

i.e.,            0 = –(200) sin θ + (50 cos 30°)

Solving, we get

For the  x  components

v PG ,  x  =  v PW, x  +  v WG, x

v PG  = (200 cos θ) + (50 sin 30°)

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Rain -man Problems

In such problems we come across the following terminology according to which

     For dealing with such like problems, we have two options

     When no wind is blowing.

v rm  =  v r  –  v m  or  v rm  =  v r  + (– v m ) 

     So, while dealing with the problems in which rain and man are there but no wind exists, then to calculate the direction of  v rm , we simply reverse the direction of man’s velocity (– v m ) and then find the resultant of – v m  and  v r  i.e., – v r  + (– v m ) to get the direction and magnitude of  v rm . It’s the Best Trick! Try following and see the results.

     When wind is blowing.

v rm  = ( v r  +  v w ) –  v m  or  v rm  =  v r  +  v w  + (– v m ) 

     So, while dealing with the problems involving rain, man and the blowing wind we first calculate the resultant of rain and wind i.e., net velocity of rain under the inference of wind  v r  +  v w  =  v net rain  –  v m . The man sees this rain falling with a velocity  v rm  =  v net rain  –  v m

     Again, we reverse the direction of velocity of man and then find the resultant of  v net rain  and  – v m  to get  v rm  with magnitude and direction.

     So, to deal with problems involving rain, man and wind we just reverse the direction of  v m  i.e., make it – v m  and then find the resultant of  v r ,  v w   and  – v m  i.e.,  v r  +  v w  + (– v m )

Relative Velocity In Two Dimensions FAQs

Ans. Motion is always in one frame of reference hence it is relative.

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