Relative velocity || relative to the at Coin appraisals

Relative velocity || relative to the

Published May 28th, 2008 in Uncategorized

Relative velocity is used in different ways:

1. The velocity of one object relative to that of another object, as measured in a single frame of reference. Alonso & Finn, Fundamental University Physics, volume 1

It is the vector subtraction of the velocities of the objects.
For example, let’s assume that the objects are two cars that move towards each other, and the reference frame is the road. The velocity of the one car (moving with velocity a) relative to the velocity of the other car (moving with velocity b) is the vector (a - b).
As this definition uses only a single frame of reference, it is applicable in both classical mechanics and special relativity http://www.fourmilab.ch/etexts/einstein/specrel/www/ par.3 .

2. The velocity of an object relative to a reference frame that can be calculated if the velocity of that object relative to another reference frame is known and the velocity of the one reference frame relative to the other reference frame is known.

For example, let’s assume that the object is an airplane, that the reference frame is the ground and that the other reference frame is the air. If the velocity of an airplane relative to the air is known (air speed and direction) and the velocity of the air relative to the ground is known (wind speed and direction), one can calculate the relative velocity of the airplane relative to the ground (ground speed and direction).

3. Relative scalar velocity: the change in distance between two objects over time.

Classical Mechanics

Scalar Velocities

Relative scalar velocity is the change in distance between two objects with respect to time. That is, if <math>s</math> is the distance between the two objects, then the relative velocity between two objects can be computed as:

<math>\mathbf{v}_{rel}=\frac{ds}{dt}</math>

Vector Velocities

In modern literature, velocity usually means vector velocity. It can be represented by a vector because it has both direction and magnitude (a number that in this case indicates speed). Calculating a relative velocity is therefore done by vector subtraction.

If an object A is moving with velocity vector v and an object B with velocity vector w , then the velocity of object A relative to object B is defined as the difference of the two velocity vectors:

<math>\mathbf{v}_{Arelative toB} = v - w</math>

Similarly the relative velocity of object B moving with velocity w, relative to object A moving with velocity v is:

<math>\mathbf{v}_{Brelative toA} = w - v</math>

Since velocity is the change in position with respect to time, two velocities, <math>v</math> and <math>w</math> could be alternatively written as the derivative of the position with respect to time:

<math>v=\frac{d\mathbf{r}_{1}}{dt}</math> , <math>w=\frac{d\mathbf{r}_{2}}{dt}</math>

For motion in only one or two dimensions it is often easier to calculate the relative velocity using the simplified equations offered by either Classical Mechanics or Special Relativity. For more complicated motion, such as three-dimensional motion, the math can be easier to calculate the rate of change of distance directly and then simply differentiate with respect to time.

In the figure above, if a is the airplane’s ground velocity and b is the velocity of the air relative to the ground, then (a-b) is the velocity of the airplane relative to the air.

In classical mechanics it does not matter if one keeps the ground as reference frame (vector subtraction) or if one switches to the air as reference frame (Galilean coordinate transformation): the outcome is the same.

With the definition of “relative velocity” as frame transformation, calculating the relative velocity by subtracting vectors is not valid when the speed component of one or both known velocities approaches the speed of light. In that case the rules of frame transformation of Special Relativity must be applied.

Special Relativity

The relativistic velocity-addition formula shows how to calculate the velocity <math>\mathbf{w}_{rel}</math> of one object in a frame S if we knew the velocity w of that object in a frame S’ and the relative velocity v between frames S and S’. If the two velocities are in the same direction:

<math>\mathbf{w}_{rel} = \frac{v+w}{1+\frac{wv}{c^2}}</math>

Using this equation, if <math>w</math> equaled c, <math>\mathbf{w}_{rel}=c</math>, so that light has the same speed in any inertial frame.