Relative Velocity
Description:
The Concept
Relative Velocity could be understood as velocity of one object with respect to the other.
There are numerous real – life experiences of the same which we can relate to, like −
- A car overtaking another car.
- Runners involved in a race, etc.
Relative Velocity – General Formula
v→AB = v→A - v→B
Where, v→AB = velocity of A with respect to B,
v→A = velocity of A with respect to ground,
v→B = velocity of B with respect to ground
E.g. consider two cars, Car A and Car B. Car B is in front of Car A and both are facing East. Now, assuming the sign convention to be,
- East (Positive)
- West (Negative)
Consider the following cases,
Case 1: Car A and Car B have same velocity with respect to ground
In this case,
A ground observer will see both the cars moving East at constant velocity.
“Car B” driver will see “Car A” in front of him always without any change in separation between them. This is because,
v→AB = v→A - v→B
Also, v→A = v→B, therefore,
v→AB = 0
Case 2: Car A has larger magnitude of velocity with respect to Car B (same direction of velocity)
In this case,
A ground observer will see both the cars moving East and “Car A” overtaking “Car B” at some point of time pertaining to its higher velocity.
“Car B” driver will see “Car A” approach him. This is because,
v→AB = v→A - v→B, Velocity of “Car A” observed by “Car B”.
The result of subtraction will be positive.
“Car B” will observe “Car A” to be moving in the East direction while approaching him.
Car A driver will see “Car B” approach him. This is because,
v→AB = v→B - v→A, Velocity of “Car A” observed by “Car B”.
The result of subtraction will be negative.
“Car A” will observe “Car B” to be moving in the West direction while approaching him.
Case 3: “Car A” has smaller magnitude of velocity with respect to “Car B” (same direction of velocity)
In this case,
A ground observer will see both the cars moving East and the separation between them increase more and more as the time passes.
“Car B” driver will see “Car A” move away from him. This is because,
v→AB = v→A - v→B, Velocity of “Car A” observed by “Car B”.
The result of subtraction will be negative.
“Car B” will observe “Car A” to be moving in the West direction while moving away.
Car A driver will see “Car B” move away from him. This is because,
v→AB = v→B - v→A, Velocity of “Car A” observed by “Car B”.
The result of subtraction will be positive.
“Car A” will observe “Car B” to be moving in the East direction while moving away.
Relative Displacement
Taking the same idea forwards, Relative displacement can be understood as displacement of one object with respect to the other.
S→AB = S→A - S→B
where, S→AB = displacement of A with respect to B,
S→A = displacement of A with respect to ground,
S→B = displacement of B with respect to ground
E.g. consider two observers standing at specific positions as shown below,
We can say,
SA = -4 units,
Also,
SB = 3 units,
However, displacement of A with respect to B will be,
SAB = SA - SB = -4 - (+3) = -7 units
7 units towards the negative direction. (shown in the figure)
Relative Acceleration
Relative acceleration can be understood as acceleration of one object with respect to the other.
a→AB = a→A - a→B
where, a→AB = displacement of A with respect to B,
where, a→A = displacement of A with respect to ground,
where, a→B = displacement of B with respect to ground