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Relative velocities between two particles in classical mechanics. The figure shows two objects A and B moving at constant velocity. The equations of motion are: = +, = +, where the subscript i refers to the initial displacement (at time t equal to zero).
In general for three objects A (e.g. Galileo on the shore), B (e.g. ship), C (e.g. falling body on ship) the velocity vector of C relative to A (velocity of falling object as Galileo sees it) is the sum of the velocity ′ of C relative to B (velocity of falling object relative to ship) plus the velocity v of B relative to A (ship's velocity ...
Denote either entity by X. Then X moves with velocity u relative to F, or equivalently with velocity u′ relative to F′, in turn F′ moves with velocity v relative to F. The inverse transformations can be obtained in a similar way, or as with position coordinates exchange u and u′, and change v to −v.
If point A has velocity components = (,,) and point B has velocity components = (,,) then the velocity of point A relative to point B is the difference between their components: / = = (,,) Alternatively, this same result could be obtained by computing the time derivative of the relative position vector r B/A .
Trajectory of a particle with initial position vector r 0 and velocity v 0, subject to constant acceleration a, all three quantities in any direction, and the position r(t) and velocity v(t) after time t. The initial position, initial velocity, and acceleration vectors need not be collinear, and the equations of motion take an almost identical ...
Then, the velocity of object A relative to object B is defined as the difference of the two velocity vectors: = Similarly, the relative velocity of object B moving with velocity w, relative to object A moving with velocity v is: = Usually, the inertial frame chosen is that in which the latter of the two mentioned objects is in rest.
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The primed frame is moving relative to the unprimed frame at velocity v. Fields defined in the primed frame are indicated by primes, and fields defined in the unprimed frame lack primes. The field components parallel to the velocity v are denoted by E ∥ and B ∥ while the field components perpendicular to v are denoted as E and B .