Velocity

Encyclopedia : V : VE : VEL : Velocity

The velocity of an object is simply its speed in a particular direction. Since velocity is defined as a vector, both speed and direction are required to define it.

Contents

1 Explanation
2 Formal description
3 Polar coordinates
4 See also

Explanation

The velocity (v) is a physical quantity of an object's motion. Velocity is speed that has direction.

The average velocity (v) of an object moving a distance (d) in a straight line during a time interval (t) is described by the formula:

[v = \frac \;].

The displacement (in the same direction as the velocity) can be found by taking the area under the graph or the integral.

Formal description

Velocity (symbol: v) is a vector measurement of the rate of change of displacement from a fixed point. The scalar absolute value (magnitude) of velocity is speed. Velocity can also be defined as rate of change of displacement or just as the rate of displacement. It is a vector quantity with dimension LT^(-1). In the SI (metric) system it is measured in metre per second.

The instantaneous velocity vector (v) of an object that has position at time (t) is given by x(t) can be computed as the derivative:

[v=x \over \mathrmt} = \lim_].

The equation for an object's velocity can be obtained mathematically by taking the integral of the equation for its acceleration beginning from some initial period time [ t_0 ] to some point in time later [t_n].

The final velocity v₂ of an object which starts with velocity v₁ and then accelerates at constant acceleration a for a period of time t is:

[v_2 = v_1 + at\;\!].

The average velocity of an object undergoing constant acceleration is [\begin \frac \; \end]. To find the displacement (d) of such an accelerating object during a time interval (t), substitute this expression into the first formula to get:

[d = t \times \frac ].

When only the object's initial velocity is known, the expression,

[d = v_1 t + \frac ],

can be used. These basic equations for final velocity and displacement can be combined to form an equation that is independent of time, also known as Torricelli's equation:

[v_2^2 = v_1^2 + 2ad].

The above equations are valid for both classical mechanics and special relativity. Where classical mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in classical mechanics, all observers agree on the value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity.

The kinetic energy (energy of motion) of a moving object is linear with both its mass and the square of its velocity:

[E_ = \begin \frac \end mv^2].

The kinetic energy is a scalar quantity.

Polar coordinates

In polar coordinates, a two-dimensional velocity can be decomposed into a radial velocity, defined as the component of velocity away from or toward the origin, and transverse velocity, the component of velocity along a circle centred at the origin, and equal to the distance to the origin times the angular velocity.

Angular momentum in scalar form is the distance to the origin times the transverse speed, or equivalently, the distance squared times the angular speed, with a plus or minus to distinguish clockwise and anti-clockwise direction.

If forces are in the radial direction only, as in the case of a gravitational orbit, angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion.

Velocity

Explanation

Formal description

Polar coordinates

See also