Velocity

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Velocity
US Navy 040501-N-1336S-037 The U.S. Navy sponsored Chevy Monte Carlo NASCAR leads a pack into turn four at California Speedway.jpg
As a change of direction occurs while the cars turn on the curved track, their velocity is not constant.
Common symbols v, v
SI unit m/s

Velocity is the rate of change of the position of an object, equivalent to a specification of its speed and direction of motion, e.g. 60 km/h to the north. Velocity is an important concept in kinematics, the branch of classical mechanics which describes the motion of bodies.

Velocity is a vector physical quantity; both magnitude and direction are required to define it. The scalar absolute value (magnitude) of velocity is called "speed", a quantity that is measured in metres per second (m/s or m·s−1) in the SI (metric) system. For example, "5 metres per second" is a scalar (not a vector), whereas "5 metres per second east" is a vector.

If there is a change in speed, direction, or both, then the object has a changing velocity and is said to be undergoing an acceleration.

Constant velocity vs acceleration[edit]

To have a constant velocity, an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path (the object's path does not curve). Thus, a constant velocity means motion in a straight line at a constant speed.

For example, a car moving at a constant 20 kilometres per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes. Hence, the car is considered to be undergoing an acceleration.

Distinction between speed and velocity[edit]

Speed describes only how fast an object is moving, whereas velocity gives both how fast and in what direction the object is moving.[1] If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified.

The big difference can be noticed when we consider movement around a circle. When something moves in a circle and returns to its starting point its average velocity is zero but its average speed is found by dividing the circumference of the circle by the time taken to move around the circle. This is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled.

Equation of motion[edit]

The average velocity \boldsymbol{\bar{v}} of an object moving through a displacement ( \Delta \boldsymbol{x}) during a time interval ( \Delta t) is described by the formula:

\boldsymbol{\bar{v}} = \frac{\Delta \boldsymbol{x}}{\Delta t}.

The velocity vector v of an object that has positions x(t) at time t and x(t + \Delta t) at time t + \Delta t, can be computed as the derivative of position:

\boldsymbol{v} = \lim_{\Delta t \to 0}{{\boldsymbol{x}(t+\Delta t)-\boldsymbol{x}(t)} \over \Delta t}={\mathrm{d} \boldsymbol{x} \over \mathrm{d}t}.

More simply, for motion in one dimension, velocity can be defined as the slope of the position vs. time graph of an object.

Average velocity magnitudes are always smaller than or equal to average speed of a given particle. Instantaneous velocity is always tangential to trajectory. Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity.

The equation for an object's velocity can be obtained mathematically by evaluating 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 v of an object which starts with velocity u and then accelerates at constant acceleration a for a period of time \Delta t is:

\boldsymbol{v} = \boldsymbol{u} + \boldsymbol{a} \Delta t.

The average velocity of an object undergoing constant acceleration is \tfrac {(\boldsymbol{u} + \boldsymbol{v})}{2}, where u is the initial velocity and v is the final velocity. To find the position, x, of such an accelerating object during a time interval, \Delta t, then:

 \Delta \boldsymbol{x} = \frac {( \boldsymbol{u} + \boldsymbol{v} )}{2}\Delta t.

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

 \Delta \boldsymbol{x} = \boldsymbol{u} \Delta t + \frac{1}{2}\boldsymbol{a} \Delta t^2,

can be used.

This can be expanded to give the position at any time t in the following way:

 \boldsymbol{x}(t) = \boldsymbol{x}(0) + \Delta \boldsymbol{x} = \boldsymbol{x}(0) + \boldsymbol{u} \Delta t  +  \frac{1}{2}\boldsymbol{a} \Delta t^2,

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

v^2 = u^2 + 2a\Delta x.\,

The above equations are valid for both Newtonian mechanics and special relativity. Where Newtonian mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in Newtonian 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. In other words only relative velocity can be calculated.

In Newtonian mechanics, the kinetic energy (energy of motion), E_K, of a moving object is linear with both its mass and the square of its velocity:

E_{K} = \begin{matrix} \frac{1}{2} \end{matrix} m \boldsymbol{v}^2.

The kinetic energy is a scalar quantity.

Escape velocity is the minimum a ballistic object needs to escape from a massive body like the earth. It represents the kinetic energy that when added to the object's gravitational potential energy (which is always negative) is greater than or equal to zero. Escape velocity from the Earth's surface is about 11 100 m/s.

Relative velocity[edit]

Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. Relative velocity is fundamental in both classical and modern physics, since many systems in physics deal with the relative motion of two or more particles. In Newtonian mechanics, the relative velocity is independent of the chosen inertial reference frame. This is not the case anymore with special relativity in which velocities depend on the choice of reference frame.

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:

\boldsymbol{v}_{A\text{ relative to }B} = \boldsymbol{v} - \boldsymbol{w}

Similarly the relative velocity of object B moving with velocity w, relative to object A moving with velocity v is:

\boldsymbol{v}_{B\text{ relative to }A} = \boldsymbol{w} - \boldsymbol{v}

Usually the inertial frame is chosen in which the latter of the two mentioned objects is in rest.

Scalar velocities[edit]

In the one-dimensional case,[2] the velocities are scalars and the equation is either:

\, v_{rel} = v - (-w), if the two objects are moving in opposite directions, or:
\, v_{rel} = v -(+w), if the two objects are moving in the same direction.

Polar coordinates[edit]

In polar coordinates, a two-dimensional velocity is described by a radial velocity, defined as the component of velocity away from or toward the origin (also known as velocity made good), and an angular velocity, which is the rate of rotation about the origin (with positive quantities representing counter-clockwise rotation and negative quantities representing clockwise rotation, in a right-handed coordinate system).

The radial and angular velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. The transverse velocity is the component of velocity along a circle centered at the origin.

\boldsymbol{v}=\boldsymbol{v}_T+\boldsymbol{v}_R

where

\boldsymbol{v}_T is the transverse velocity
\boldsymbol{v}_R is the radial velocity.

The magnitude of the radial velocity is the dot product of the velocity vector and the unit vector in the direction of the displacement.

v_R=\frac{\boldsymbol{v} \cdot \boldsymbol{r}}{\left|\boldsymbol{r}\right|}

where

\boldsymbol{r} is displacement.

The magnitude of the transverse velocity is that of the cross product of the unit vector in the direction of the displacement and the velocity vector. It is also the product of the angular speed \omega and the magnitude of the displacement.

v_T=\frac{|\boldsymbol{r}\times\boldsymbol{v}|}{|\boldsymbol{r}|}=\omega|\boldsymbol{r}|

such that

\omega=\frac{|\boldsymbol{r}\times\boldsymbol{v}|}{|\boldsymbol{r}|^2}.

Angular momentum in scalar form is the mass times the distance to the origin times the transverse velocity, or equivalently, the mass times the distance squared times the angular speed. The sign convention for angular momentum is the same as that for angular velocity.

L=mrv_T=mr^2\omega\,

where

m\, is mass
r=\|\boldsymbol{r}\|.

The expression mr^2 is known as moment of inertia. If forces are in the radial direction only with an inverse square dependence, 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.

See also[edit]

Notes[edit]

  1. ^ Wilson, Edwin Bidwell (1901). Vector analysis: a text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs. p. 125.  This is the likely origin of the speed/velocity terminology in vector physics.
  2. ^ Basic principle

References[edit]

  • Robert Resnick and Jearl Walker, Fundamentals of Physics, Wiley; 7 Sub edition (June 16, 2004). ISBN 0-471-23231-9.

External links[edit]