Define addition of velocities & formula
Sometimes an object moves inside another moving object.
For example, a boy walks inside a moving train or a swimmer swims in flowing water.
In such situations, the final speed observed from the ground changes because both motions combine together.
This combined motion is called Addition of Velocities.
A velocity-addition formula is a 3-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different frames. The speed of light in the fluid is slower than the speed of light in vacuum, and it changes if the fluid is moving along with the light.
In 1851, Fizeau measured the speed of light in a fluid moving parallel to the light using a Michelson interferometer. Fizeau's results were not in accord with the then prevalent theories.
Fizeau experimentally correctly determined the zeroth term of an expansion of the relativistically correct addition law in terms of V/c .
Understanding Relative Velocity
Velocity always depends on the observer. A person sitting inside a train may appear stationary to another passenger, but the same person appears moving to someone standing on the ground.
Relativistic velocity addition formula
u = (v+w) /(1 + (v * w)/c²)- where:
- u is speed of the projectile when outside of the spaceship
- v is the speed of the spaceship
- w is the speed of the projectile when we see from the spaceship
- c is the speed of light
The Einstein velocity relationship transforms a measured velocity as seen in one inertial frame of reference (u) to the velocity as measured in a frame moving at velocity v with respect to it (u').
When Velocities Are in Opposite Directions
If both objects move in opposite directions, the velocities are subtracted.
V = u - v
Daily Life Examples of Addition of Velocities
- A person walking inside a moving bus
- A boat moving in flowing river water
- An airplane flying with wind
- A runner moving on an escalator
Standard applications of velocity-addition formulas
- 1. Aberration of light
- 2. Doppler shift
- 3. Doppler navigation
Velocity is a vector quantity, so the relationships between the magnitude, direction, x-axis component and y-axis component are important.
These vector components can be added analytically or graphically.
Difference Between Classical and Relativistic Addition
| Classical Addition | Relativistic Addition |
|---|---|
| Used for normal speeds | Used near speed of light |
| Simple addition/subtraction | Uses Einstein formula |
| Can exceed light speed mathematically | Never exceeds speed of light |
Solved Example
A train moves with a velocity of 20 m/s. A boy runs inside the train in the same direction with velocity 5 m/s. Find the velocity of the boy relative to the ground.
Using:
V = u + v
V = 20 + 5 = 25 m/s
Therefore, the velocity of the boy relative to the ground is 25 m/s.
Quick Revision
- Velocities add in same direction
- Velocities subtract in opposite direction
- Relative velocity depends on observer
- Relativity is used near light speed
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