This belongs to the Relative Velocity of undergraduate physics.

1. Relative Velocity
UK Physics-9841895036
Department of PHYSICS
CCRC
2077
1 Defination
The relative velocity of a body A, moving with velocity
~
VA with respect to the another body B, moving with
the velocity ~
VB,is defined as: the velocity with which the
body A appears to move when the body B is brought into
rest. It is denoted by ~
VAB, read as; the relative velocity
of A with respect to B, and is expressed as;
~
VAB = ~
VA − ~
VB (1)
Here, the observer is brought in rest and measures the
apparent velocity of others. In the vector diagram, rest
is approached by taking the reverse direction of the ob-
server.
1.1 In one dimension
1.1.1 In the same direction
~
VAB = ~
VA − ~
VB = (5 − 3)m/s = 2m/s
and,
~
VBA = ~
VB − ~
VA = (3 − 5)m/s = −2m/s
Here, the negative sign indicates the transformation of
velocity.
1.1.2 In the opposite direction
~
VAB = ~
VA − (−~
VB) = ~
VA + ~
VB = (5 + 3)m/s = 8m/s
and,
~
VBA = ~
VB − (−~
VA) = ~
VB + ~
VA = (3 + 5)m/s
= 8m/s
Here in both cases, the bodies approach fast.
1.2 In two dimensions
Consider two bodies A and B with velocity vectors ~
VA
and ~
VB.
The relative velocity of the body A with respect to the
body B is;
~
VAB = ~
VA − ~
VB
Here, the observer B in motion with ~
VB, finds the
body A in motion with ~
VA, apparently moving along
the direction of ~
VAB.
In this case, the magnitude and direction of the
resultant will be;
1

2. |~
VAB| =
q
|~
VA|2 + |~
VB|2 − 2|~
VA||~
VB| cos θ
and: α = tan−1
|~
VB| sin θ
|~
VA|−|~
VB| cos θ









1
and,
the relative velocity of the body B with respect to
the body A is;
~
VBA = ~
VB − ~
VA
Here, the observer A in motion with ~
VA, finds the
body B in motion with ~
VB, apparently moving along the
direction of ~
VBA
In this case, the magnitude and direction of the
resultant will be;
|~
VBA| =
q
|~
VB|2 + |~
VA|2 − 2|~
VB||~
VA| cos θ
and, : α = tan−1
|~
VA| sin θ
|~
VB|−|~
VA| cos θ









2
Homework
Solve the above equations, 1 and 2 for different
cases;
i) for θ = 0◦
ii) for θ = 90◦
iii) for θ = 180◦
iv) for |~
VA| = |~
VB|
2 Relative velocity of rain w.r.t.
man
Consider a case in which rain is falling vertically down-
ward with ~
Vr and man is moving towards east with ~
Vm,
as in the figure.
Figure 1: Actual direction of rain and man
Here,
The relative velocity of rain w.r.t. man is denoted
by ~
Vrm and is expressed as;
~
Vrm = ~
Vr − ~
Vm.
In the vector diagram, this relation can be repre-
sented as;
Figure 2: Relative velocity of rain w.r.t. man
From the figure, we have;
Magnitude: |~
Vrm| =
q
|~
Vr|2 + |~
Vm|2
and direction : θ◦
= tan−1
|~
Vm|
|~
Vr|
Thus, the man holding an umbrella should tilt his
umbrella at an angle of θ◦
with the vertical so as to
protect himself from the rain.
The rain appears falling with some oblique angle
θ◦
as in the figure.
Example
A man, walking on a road with a velocity of 5
km/hr, encounters rain falling vertically downwards
with a velocity of 12 km/hr. What is the magnitude
of the relative velocity of rain with respect to man?
2

3. At what angle should he tilt his umbrella in order to
protect himself from the rain?
Solution,
Figure 3: Vector diagram
From the figure, we have;
Magnitude of the relative velocity of rain w.r.t.
man is given by:
|~
Vrm| =
q
|~
Vr|2 + |~
Vm|2
here,
~
Vr = 12km/hr
and ~
Vm = 5km/hr.
With these values, we will get
|~
Vrm| = 13km/hr
and direction : θ = tan−1
|~
Vm|
|~
Vr|
Using the values, we will get, θ = 22.6◦
Thus the man should he tilt his umbrella at an
angle of 22.6◦
with the vertical, in order to protect
himself from the rain
3 Crossing river problem
Consider a river current has a velocity ~
u and a man can
swim in a still water with a velocity ~
v.
For shortest time
The time required to cross from A to B can be
Figure 4: Crossing river problem
expressed as;
T = distance
V elocitycomponentalongAB
or, T = d
v cos θ
For shortest time, cos θ should be maximum. So,
cos θ = 1, =⇒ θ = 0◦
∴ Tshortest = d
v .
Thus, for the shortest time, the swimmer should
swim perpendicular to the flow.
For shortest path
The distance AB = d is the shortest path. For
shortest path, the horizontal component should balance
the speed of the flow of river.
i.e., v sin θ = u
or, sin θ = u
v
or, θ = sin−1
(u
v ).
Thus, to cross the river along the shortest path,
the man should inclined to the left at an angle of
sin−1
(u
v ) with perpendicular to the flow.
Example
A man who can swim in still water with a speed
of 2 m/s wants to cross a river 200 m broad, flowing at
a speed of 1 m/s. Find, (i) the time of crossing by the
shortest path, (ii) the minimum time of crossing.
Solution.
For the first case,
From the figure, the time of crossing by the short-
est path is given by;
3

4. Figure 5: Crossing river problem
T = distance
V elocitycomponentalongAB,(VAB) · · · 1
Here, distance = 200m and velocity component along AB
is given by VAB =
√
AC2 − AB2 =
√
22 − 12 = 1.73m/s.
Finally, using these values in equation 1, we will
get, T=115.6 s.
For the second case,
The minimum time of crossing is given by;
Tshortest = d
v .
Using, d = 200m, and v = 2m/s, finally we will
get, T=100 s.
Some solved questions
1. A car is driving down the road at a velocity
Vc, relative to ground, and is delivering newspapers to
homes, as shown in the figure below. The newspapers
are thrown at a velocity of Vp relative to the car. At
what angle θ must the newspapers be thrown, relative
to the car, so that they fly in a direction parallel to the
driveways?
Solution,
For the newspapers to fly parallel to the driveways
when they are airborne, the component of velocity
of Vp that is parallel to the road must cancel out
with Vc, so that the velocity of the newspaper with re-
spect to ground is in a direction parallel to the driveways.
Using trigonometry,
Vp sin θ = Vc, and θ = sin−1
Vc
Vp
.
Homeworks
Level - 1
1. A car is going east at 60 km/hr. A truck is
going south at 30 km/hr. What is the velocity the truck
with respect to the car? [Ans: 67 km/hr at 64.4◦
West
of South]
2. A truck travelling at a 50 km/hr towards north
turns west at the same speed. What is the change in
velocity? [Ans: 50
√
2 at South West]
3. A particle posses two velocities 10 m/s and 2
m/s inclined at an angle of 120◦
. Find the magnitude
and direction of the resultant velocities. [Ans: magni-
tude; 9.17 m/s and direction; 10.9◦
]
4. A stone attached to a string is whirled round
in a horizontal circle with a constant speed of 10 m/s.
Calculate the difference in the velocity when the stone
is (i) at the opposite ends of a diameter (ii) in two
positions of A and B, where angle AOB is 90◦
and O is
the center of the circle. [Ans: (i); 20 m/s, (ii); 14.1 m/s]
5. A swimmer swimming across a river flowing at
a velocity of 4 m/s swims at the velocity of 2 m/s.
Calculate the actual velocity of the swimmer and the
angle. [Ans: 4.47 m/s and 26.57◦
]
6. The president’s airplane, Air Force One, flies at
250 m/s to the east with respect to the air. The air
is moving at 35 m/s to the north with respect to the
ground. Find the velocity of Air Force One with respect
to the ground. [Ans: 252 m/s, 8◦
North of East]
Level - 2
1. A boat is rowed with a velocity of 3 m/s straight
across a river which flows at the rate of 1 m/s. If the
river is 100 m wide, find how many far down the river,
the boat will reach the opposite bank below the point at
4

5. which it was originally directed. [Ans: far distance; 33.3
m]
2. A stream runs with a velocity of 1 m/s, find in
what direction a swimmer, whose velocity is 1.5 m/s,
should start in order to cross the stream perpendic-
ularly. What direction should he take if he wants to
cross the stream in the shortest time? [Ans: first case;
41.8◦
, second case; perpendicular to the stream’s current
3. An airplane pilot wishes to fly due north. A
wind 30 km/hr is blowing towards the west. If the
flying speed of the plane is 360 km/hr, in what direction
should the pilot set his course? What is the speed of
the plane over the ground? [Ans: first case; 4.8◦
, second
case; 359.6 km/hr]
4. In flying an airplane, a pilot wants to attain a
velocity of 500 km/hr eastward with respect to the
ground. A wind is blowing southwards at 40 km/hr.
What velocity must the pilot maintain with respect to
the air to achieve the desired ground velocity? [Ans:
502 km/hr at 4.57◦
North of East]
5. To a person going due east in a car with a ve-
locity of 25 km/hr, a train appears to move due north
with a velocity of 25
√
3 km/hr.What is the actual
velocity and direction of motion of the train? [Ans; 50
km/hr at 30◦
East of North]
Level - 3
1. For a cyclist running at 5 m/s due west, the
rain appears to fall vertically. When he increases his
speed to 17 m/s due west, the rain appears to fall at an
angle of 45◦
with the vertical. Find the actual speed
and the direction of the rain. [Ans: 13 m/s at 22.6◦
with the vertical]
2. Two ships A and B are 4 km apart. A is due
west of B. If A moves with a uniform velocity of 8
km/hr due east and B moves with a uniform velocity of
6 km/hr due south, calculate, (i) the magnitude of the
velocity of A relative to B (ii) the closest distance apart
of A and B. [Ans: (i); 10 km/hr at 36.9◦
at North of
East, second part; 2.4 km]
3. A scotch yoke mechanism is shown in the fig-
ure below, with dimensions given in units of meters.
The pin is rigidly connected to the crank wheel which is
rotating at w radians per second. Since the scotch yoke
is being used in a food plant and contamination must
be prevented, the pin is made of smooth plastic, with
no lubrication applied. To prevent excessive frictional
heating, the sliding velocity of the pin relative to the
slot is to be no more than 5 m/s. What is the maximum
value of w to make sure this limit is not exceeded? [Ans:
w = 5
( d
2 +R)
.]
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