Relative velocity
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Relative velocity is an essential concept in both classical and modern physics, since nearly all significant problems in physics deal with the relative velocity of two or more bodies. It is especially important in special relativity.
The velocity of any object in motion can only be measured relative to either some other object, or to a frame of reference. Say an object is placed on a table, and is at rest. You would describe it as being at rest even though you know very well that the table is on the floor, which is on the surface of the Earth, which is rotating about its axis as well as revolving around the sun. The reason for this is that you will have measured the velocity of the object relative to the table, and relative to the table, its velocity is indeed zero.
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[edit] Introduction
The expression relative velocity is used in different ways:
1. The velocity of one object relative to another object, as measured in a single reference frame.[1]
For one-dimensional problems it simply is the difference in velocity; in general one has to subtract the velocity of one from that of the other.
For example, let's assume that two cars move towards each other. The velocity of car A (moving with velocity vA) relative to car B (moving with velocity vB) is the vector (vA - vB). If car A moves with 100 km/h to the right (relative to the road of course!), and car B with 100 km/h to the left, the velocity of A is +100 and the velocity of B is -100 km/h(taking B's velocity with respect to right,i.e.,with respect to the direction of velocity of A); then the velocity of A relative to B is 200 km/h and, inversely, the velocity of B relative to A is -200 km/h.
- Here the velocities vA and vB are defined relative to the same frame of reference; the resulting value is valid for classical mechanics as well as for special relativity [2].
2. The velocity relative to a moving body, as measured in a co-moving frame of reference [3].
For example, let's assume that you are in an airplane and you want to know how fast the airplane flies. If the velocity of the airplane relative to the ground is known (ground speed and direction, as determined in the control tower) as well as the velocity of the air relative to the ground (wind speed and direction), one can calculate the velocity of the airplane relative to the air (air speed and direction, as measured with the air as reference frame).
- This corresponds to a coordinate transformation to the reference frame of the moving body.
3. The change in distance between two objects over time[citation needed], such as used in the Doppler effect.
For example, a geostationary satellite can be said to be "stationary" relative to a ground antenna according to the third usage.
Such a satellite is also stationary relative to the rotating Earth, but it is orbiting in the Earth centered inertial frame. In classical mechanics as well as special relativity, inertial frames are preferred for calculations of motion.
In most modern physics literature, velocities are vectors, not scalars. For scalar velocity the word "speed" is preferred.
[edit] Classical Mechanics
[edit] Scalar Velocities
"Relative scalar velocity" is sometimes used[citation needed] for 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>
[edit] Vector Velocities
In modern literature, velocity usually means vector velocity. It can be represented by a vector because it has both direction and magnitude (speed in this case). 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.
[edit] Special Relativity
See main article velocity-addition formula.
The relativistic velocity-addition formula shows how to calculate the velocity <math>\mathbf{w}_{rel}</math> of one object in a frame S if the velocity w of that object in a frame S' is known and the relative velocity v between frames S and S' is known. 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.
[edit] References
- ^ Alonso & Finn, Fundamental University Physics, volume 1
- ^ A. Einstein, 1905 http://www.fourmilab.ch/etexts/einstein/specrel/www/ par.3
- ^ H. Benson, University Physics. Wiley&Sons, 1991
[edit] See also
[edit] External links
- math.ucr
- Hyperphysics.phyes:Velocidad relativa
fr:Vitesse relative ko:속도 ru:Теорема о сложении скоростей
