Relative velocity

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Relative velocity is a measurement of velocity between two objects moving in different frames of reference. Relative velocity is an essential area of both classical and modern physics, since nearly all non-trivial problems in physics deal with the relative velocity of two or more particles. It is especially important in special relativity where there is no such thing as absolute motion, thus making all motion and therefore velocities relative. Since velocity is the change in position with respect to time, two velocities, $v$ and $w$ could be alternatively written as the derivative of the position with respect to time:

$v=\frac{d\mathbf{r}_{1}}{dt}$ , $w=\frac{d\mathbf{r}_{2}}{dt}$

Relative velocity is said to be in the classical range, that is it is determined by Classical mechanics, when, if we let c be defined as the Speed of light in a Vacuum:

$\frac{v+w}{c}$ ≈

0

or almost equivalently,

$\frac{v*w}{c^2}$ ≈

0

If it is not approximately zero, then the speeds are considered to be in the relativistic range, that is it is determined by Special relativity.

Fundamentally, regardless of whether the velocities are classical or relativistic, relative velocity is the change in distance between the two objects with respect to time. That is, if $s$ is the distance between the two objects, then the relative velocity between two objects can be computed exactly as:

$\mathbf{v}_{rel}=\frac{ds}{dt}$

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.

1 Classical Mechanics
- 1.1 Vector Velocities
- 1.2 Scalar Velocities
2 Special Relativity
3 See also

[edit] Classical Mechanics

Within the realm of classical mechanics, relative velocity is a very basic concept which can be calculated using either vector or scalar notation.

[edit] Vector Velocities

According to classical mechanics, if an object A is moving with velocity vector v and an object B with velocity vector w, the velocity of object A relative to object B is defined as the addition of the two velocity vectors:

$\mathbf{v}_{rel} = v - w$

[edit] Scalar Velocities

In the one dimensional case, the velocities are scalars and the equation is either:

$\mathbf{v}_{rel} = v - (-w)$ , if the two objects are moving in opposite directions, or:

$\mathbf{v}_{rel} = v -(+w)$ , if the two objects are moving in the same direction.

In two or three dimensions it is essential to utilise vector notation as the motion will have components in each dimension, easily handled in vector notation.

[edit] Special Relativity

[edit] Velocity Constraint

The concept of relative velocity in special relativity has a complication that does not come into the classical calculation. That complication is the requirement that both velocities $v$ and $w$ are constants. This is because special relativity is only valid for non-accelerated motion. This means that $v$ and $w$ are still:

$v=\frac{d\mathbf{r}_{1}}{dt}$ , $w=\frac{d\mathbf{r}_{2}}{dt}$

but now the additional constraint is added, which will also give us that:

$\mathbf{a}_{1}=\frac{dv}{dt}=0$ , $\mathbf{a}_{2}=\frac{dw}{dt}=0$

where $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ are the respective accelerations.

[edit] Relativistic Range

Special relativity states that the speed of light can not be exceeded. According to classical mechanics, if two objects moving away from each other each with a velocity of c would have a relative velocity of 2c. In special relativity, the speed 2c is not possible. Therefore, to calculate the relative velocity between two objects, one must use the relativistic equation:

$\mathbf{v}_{rel} = \frac{w-v}{1-\frac{w*v}{c^2}}$

Using this equation, even if both $v$ and $w$ equaled c, the addition of the two would also equal c.

[edit] Derivation of Classical Equation

The derivation of the classical equation for relative velocities is a very simple derivation, combining only one equation and one generalization already stated above. Starting with the special relativity equation for relative velocity, and applying the second approximation given for determining the classical range, we get:

Assuming the following approximation:

$\frac{v*w}{c^2}$ ≈

0

our equation can be approximated to:

$\mathbf{v}_{rel}$	$= \frac{\mathbf{w}-\mathbf{v}}{1-\frac{\mathbf{w}*\mathbf{v}}{\mathbf{c}^2}}$
	$= \frac{\mathbf{w}-\mathbf{v}}{1-0}$
	$= \frac{\mathbf{w}-\mathbf{v}}{1}$
	$= \mathbf{w-v}$