laws of motion are valid.).
The observers themselves are not accelerated. There is, however, no restriction on the motion of
the object itself, which the observers are going to observe from different reference systems. The
motion of the object can very well be accelerated. Further, we shall study relative motion for two
categories of motion : (i) one dimension (in this module) and (ii) two dimensions (in another
module). We shall skip three dimensional motion – though two dimensional study can easily be
extended to three dimensional motion as well.
Relative motion in one dimension
We start here with relative motion in one dimension. It means that the individual motions of the
object and observers are along a straight line with only two possible directions of motion.
Position of the point object
We consider two observers “A” and “B”. The observer “A” is at rest with earth, whereas observer
“B” moves with a velocity v B A with respect to the observer “A”. The two observers watch the
motion of the point like object “C”. The motions of “B” and “C” are along the same straight line.
It helps to have a convention about writing subscripted symbol such as v B A . The first
subscript indicates the entity possessing the attribute (here velocity) and second subscript
indicates the entity with respect to which measurement is made. A velocity like v B A shall,
therefore, mean velocity of “B” with respect to “A”.
The position of the object “C” as measured by the two observers “A” and “B” are x C A and x C B as shown in the figure. The observers are represented by their respective frame of reference in the
figure.
Figure 3.1. Position
Here,
Velocity of the point object
We can obtain velocity of the object by differentiating its position with respect to time. As the
measurements of position in two references are different, it is expected that velocities in two
references are different, because one observer is at rest, whereas other observer is moving with
constant velocity.
and
Now, we can obtain relation between these two velocities, using the relation
x C A = x B A + x C B and differentiating the terms of the equation with respect to time : Figure 3.2. Relative velocity
The meaning of the subscripted velocities are :
v C A : velocity of object "C" with respect to "A"
v C B : velocity of object "C" with respect to "B"
v B A : velocity of object "B" with respect to "A"
Example 3.1.
Problem : Two cars, standing a distance apart, start moving towards each other with speeds 1
m/s and 2 m/s along a straight road. What is the speed with which they approach each other ?
Solution : Let us consider that "A" denotes Earth, "B" denotes first car and "C" denotes second car. The equation of relative velocity for this case is :
Here, we need to fix a reference direction to assign sign to the velocities as they are moving
opposite to each other and should have opposite signs. Let us consider that the direction of the
velocity of B is in the reference direction, then
Figure 3.3. Relative velocity
Now :
This means that the car "C" is approaching "B" with a speed of -3 m/s along the straight road.
Equivalently, it means that the car "B" is approaching "C" with a speed of 3 m/s along the
straight road. We, therefore, say that the two cars approach each other with a relative speed of
3 m/s.
Acceleration of the point object
If the object being observed is accelerated, then its acceleration is obtained by the time derivative
of velocity. Differentiating equation of relative velocity, we have :
The meaning of the subscripted accelerations are :
a C A : acceleration of object "C" with respect to "A"
a C B : acceleration of object "C" with respect to "B"
a B A : acceleration of object "B" with respect to "A"
But we have restricted ourselves to reference systems which are moving at constant velocity. This
means that relative velocity of "B" with respect to "A" is a constant. In other words, the
acceleration of "B" with respect to "A" is zero i.e. a B A = 0 . Hence,
The observers moving at constant velocity, therefore, measure same acceleration of the object. As
a matter of fact, this result is characteristics of inertial frame of reference. The reference frames,
which measure same acceleration of an object, are inertial frames of reference.
Interpretation of the equation of relative velocity
The important aspect of relative velocity in one dimension is that velocity has only two possible
directions. We need not use vector notation to write or evaluate equation of relative velocities in
one dimension. The velocity, therefore, can be treated as signed scalar variable; plus sign
indicating velocity in the reference direction and minus sign indicating velocity in opposite to the
reference direction.
Equation with reference to earth
The equation of relative velocities refers velocities in relation to different reference system.
We note that two of the velocities are referred to A. In case, “A” denotes Earth’s reference, then
we can conveniently drop the reference. A velocity without reference to any frame shall then mean
Earth’s frame of reference.
This is an important relation. This is the working relation for relative motion in one dimension.
We shall be using this form of equation most of the time, while working with problems in relative
motion. This equation can be used effectively to determine relative velocity of two moving
objects with uniform velocities (C and B), when their velocities in Earth’s reference are known.
Let us work out an exercise, using new notation and see the ease of working.
Example 3.2.
Problem : Two cars, initially 100 m distant apart, start moving towards each other with
speeds 1 m/s and 2 m/s along a straight road. When would they meet ?
Solution : The relative velocity of two cars (say 1 and 2) is :
Let us consider that the direction v 1 is the positive reference direction.
Here, v 1 = 1 m/s and v 2 = -2 m/s. Thus, relative velocity of two cars (of 2 w.r.t 1) is :
This means that car "2" is approaching car "1" with a speed of -3 m/s along the straight road.
Similarly, car "1" is approaching car "2" with a speed of 3 m/s along the straight road.
Therefore, we can say that two cars are approaching at a speed of 3 m/s. Now, let the two cars
meet after time “t” :
Order of subscript
There is slight possibility of misunderstanding or confusion as a result of the order of subscript in
the equation. However, if we observe the subscript in the equation, it is easy to formulate a rule as
far as writing subscript in the equation for relative motion is concerned. For any two subscripts
say “A” and “B”, the relative velocity of “A” (first subscript) with respect to “B” (second
subscript) is equal to velocity of “A” (first subscript) subtracted by the velocity of “B” (second
subscript) :
and the relative velocity of B (first subscript) with respect to A (second subscript) is equal to
velocity of B (first subscript) subtracted by the velocity of A (second subscript):
Evaluating relative velocity by making reference object stationary
An inspection of the equation of relative velocity points to an interesting feature of the equation.
We need to emphasize that the equation of relative velocity is essentially a vector equation. In one
dimensional motion, we have taken the liberty to write them as scalar equation :
Now, the equation comprises of two vector quantities ( v B and – v A ) on the right hand side of the
equation. The vector “ – v A ” is actually the negative vector i.e. a vector equal in magnitude, but
opposite in direction to “ v A ”. Thus, we can evaluate relative velocity as following :
1. Apply velocity of the reference object (say object "A") to both objects and render the reference
object at rest.
2. The resultant velocity of the other object ("B") is equal to relative velocity of "B" with respect to "A".
This concept of rendering the reference object stationary is explained in the figure below. In order
to determine relative velocity of car "B" with reference to car "A", we apply velocity vector of car
"A" to both cars. The relative velocity of car "B" with respect to car "A" is equal to the resultant velocity of car "B".
Figure 3.4. Relative velocity
This technique is a very useful tool for consideration of relative motion in two dimensions.
Direction of relative velocities
For a pair of two moving objects moving uniformly, there are two values of relative velocity
corresponding to two reference frames. The values differ only in sign – not in magnitude. This is
clear from the example here.
Example 3.3.
Problem : Two cars start moving away from each other with speeds 1 m/s and 2 m/s along a
straight road. What are relative velocities ? Discuss the significance of their sign.
Solution : Let the cars be denoted by subscripts “1” and “2”. Let us also consider that the
direction v 2 is the positive reference direction, then relative velocities are :
Figure 3.5. Relative velocity
The sign attached to relative velocity indicates the direction of relative velocity with respect
to reference direction. The directions of relative velocity are different, depending on the
reference object.
However, two relative velocities with different directions mean same physical situation. Let
us read the negative value first. It means that car 1 moves away from car 2 at a speed of 3 m/s
in the direction opposite to that of car 2. This is exactly the physical situation. Now for
positive value of relative velocity, the value reads as car 2 moves from car 1 in the direction
of its own velocity. This also is exactly the physical situation. There is no contradiction as far
as physical interpretation is concerned. Importantly, the magnitude of approach – whatever be
the sign of relative velocity – is same.
Figure 3.6. Relative velocity
Relative velocity .vs. difference in velocities
It is very important to understand that relative velocity refers to two moving bodies – not a single
body. Also that relative velocity is a different concept than the concept of "difference of two
velocities", which may pertain to the same or different objects. The difference in velocities
represents difference of “final” velocity and “initial” velocity and is independent of any order of
subscript. In the case of relative velocity, the order of subscripts are important. The expression for
two concepts viz relative velocity and difference in velocities may look similar, but they are
different concepts.
Relative acceleration
We had restricted out discussion up to this point for objects, which moved with constant velocity.
The question, now, is whether we can extend the concept of relative velocity to acceleration as
well. The answer is yes. We can attach similar meaning to most of the quantities - scalar and
vector both. It all depends on attaching physical meaning to the relative concept with respect to a
particular quantity. For example, we measure potential energy (a scalar quantity) with respect to
an assumed datum.
Extending concept of relative velocity to acceleration is done with the restriction that
measurements of individual accelerations are made from the same reference.
If two objects are moving with different accelerations in one dimension, then the relative
acceleration is equal to the net acceleration following the same working relation as that for
relative velocity. For example, let us consider than an object designated as "1" moves with
acceleration " a 1 " and the other object designated as "2" moves with acceleration " a 2 " along a straight line. Then, relative acceleration of "1" with respect to "2" is given by :
Similarly,relative acceleration of "2" with respect to "1" is given by :
Worked out problems
Example 3.4. Relative motion
Problem :Two trains are running on parallel straight tracks in the same direction. The train,
moving with the speed of 30 m/s overtakes the train ahead, which is moving with the speed of
20 m/s. If the train lengths are 200 m each, then find the time elapsed and the ground distance
covered by the trains during overtake.
Solution : First train, moving with the speed of 30 m/s overtakes the second train, moving
with the speed of 20 m/s. The relative speed with which first train overtakes the second train,
The figure here shows the initial situation, when faster train begins to overtake and the final
situation, when faster train goes past the slower train. The total distance to be covered is equal
to the sum of each length of the trains (L1 + L2) i.e. 200 + 200 = 400 m. Thus, time taken to
overtake is :
Figure 3.7. The total relative distance
The total relative distance to cover during overtake is equal to the sum of lengths of each train.
In this time interval, the two trains cover the ground distance given by:
In the question given in the example, if the trains travel in the opposite direction, then find the
time elapsed and the ground distance covered by the trains during the period in which they cross
each other.
The total distance to be covered is equal to the sum of each length of the trains i.e. 200 + 200 =
400 m. Thus, time taken to overtake is :
Now, in this time interval, the two trains cover the ground distance given by:
In this case, we find that the sum of the lengths of the trains is equal to the ground distance
covered by the trains, while crossing each other.
Check your understanding
Check the module titled Relative velocity in one dimension (Check your understanding) to test your understanding of the topics covered in this module.
3.2. Relative velocity in one dimension(application)*
Questions and their answers are presented here in the module text format as if it were an extension
of the treatment of the topic. The idea is to provide a verbose explanation, detailing the
application of theory. Solution presented is, therefore, treated as the part of the understanding
process – not merely a Q/A session. The emphasis is to enforce ideas and concepts, which can not
be completely absorbed unless they are put to real time situation.
Hints on solving problems
1. Foremost thing in solving problems of relative motion is about visualizing measurement. If we
say a body "A" has relative velocity "v" with respect to another moving body "B", then we
simply mean that we are making measurement from the moving frame (reference) of "B".
2. The concept of relative velocity applies to two objects. It is always intuitive to designate one
of the objects as moving and other as reference object.
3. It is helpful in solving problem to make reference object stationary by applying negative of its
velocity to both objects. The resultant velocity of the moving object is equal to the relative
velocity of the moving object with respect to reference object. If we interpret relative velocity
in this manner, it gives easy visualization as we are accustomed to observing motion from
stationary state.
Representative problems and their solutions
We discuss problems, which highlight certain aspects of the study leading to the relative velocity
in one dimension.
Example 3.5.
Problem : A jet cruising at a speed of 1000 km/hr ejects hot air in the opposite direction. If
the speed of hot air with respect to Jet is 800 km/hr, then find its speed with respect to ground.
Solution : Let the direction of Jet be x – direction. Also, let us denote jet with “A” and hot air
with “B”. Here,
Now,
The speed of the hot air with respect to ground is 200 km/hr.
Example 3.6.
Problem : If two bodies, at constant speeds, move towards each other, then the linear distance
between them decreases at 6 km/hr. If the bodies move in the same direction with same
speeds, then the linear distance between them increases at 2 km/hr. Find the speeds of two
bodies (in km/hr).
Solution : Let the speeds of the bodies are “u” and “v” respectively. When they move towards
each other, the relative velocity between them is :
When they are moving in the same direction, the relative velocity between them is :
Solving two linear equations, we have :
u = 4 km/hr
v = 2 km/hr
Example 3.7.
Problem : Two cars A and B move along parallel paths from a common point in a given
direction. If “u” and “v” be their speeds (u > v), then find the separation between them after
time “t”.
Solution : The relative velocity of the cars is :
The separation between the cars is :
Example 3.8.
Problem : Two trains of length 100 m each, running on parallel track, take 20 seconds to
overtake and 10 seconds to cross each other. Find their speeds (in m/s).
Solution : The distance traveled in the two cases is 100 + 100 = 200m.
Let their speeds be “u” and “v”. Now, relative velocity for the overtake is :
The relative velocity to cross each other is :
Solving two linear equations, we have :
u = 15 m/s and v = 5 m/s.
Example 3.9.
Problem : Two cars are moving in the same direction at the same speed,”u”. The cars
maintain a linear distance "x" between them. An another car (third car) coming from opposite
direction meets the two cars at an interval of “t”, then find the speed of the third car.
Solution : Let “u” be the speed of either of the two cars and “v” be the speed of the third car.
The relative velocity of third car with respect to either of the two cars is :
The distance between the two cars remains same as they are moving at equal speeds in the
same direction. The events of meeting two cars separated by a distance “x” is described in the
context of relative velocity. We say equivalently that the third car is moving with relative
velocity "u+v", whereas the first two cars are stationary. Thus, distance covered by the third
car with relative velocity is given by :
Example 3.10.
Problem : Two cars, initially at a separation of 12 m, start simultaneously. First car “A”,
starting from rest, moves with an acceleration 2 m / s 2 , whereas the car “B”, which is ahead,
moves with a constant velocity 1 m/s along the same direction. Find the time when car “A”
overtakes car “B”.
Solution : We shall attempt this question, using concept of relative velocity. First we need to
understand what does the event “overtaking” mean? Simply said, it is the event when both cars
