Welcome to our website!

Position and its derivatives

The velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time or

In classical mechanics, velocities are directly additive and subtractive. For example, if one car travelling East at 60 km/h passes another car travelling East at 50 km/h, from the perspective of the car it passes it is travelling East at 60−50 = 10 km/h. From the perspective of the faster car, the slower car is moving 10 km/h to the West. What if the car is travelling north? Velocities are directly additive as vector quantities; they must be dealt with using vector analysis.

Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector v = vd and the velocity of the second object by the vector u = ue where v is the speed of the first object, u is the speed of the second object, and d and e are unit vectors in the directions of motion of each particle respectively, then the velocity of the first object as seen by the second object is:

v' = v - u

Similarly:

u' = u - v

When both objects are moving in the same direction, this equation can be simplified to:

v' = ( v - u ) d

Or, by ignoring direction, the difference can be given in terms of speed only:

v' = v - u