Speed and Velocity

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Speed is the rate of motion, or equivalently the rate of change in position, many times expressed as distance d traveled per unit of time t.

Speed is a scalar quantity with dimensions distance/time; the equivalent vector quantity to speed is known as velocity. Speed is measured in the same physical units of measurement as velocity, but does not contain the element of direction that velocity has. Speed is thus the magnitude component of velocity.

In mathematical notation, it is simply:

Objects that move horizontally as well as vertically (such as aircraft) distinguish forward speed and climbing speed.

Units

Units of speed include:

  • meters per second, (symbol m/s), the SI derived unit
  • kilometers per hour, (symbol km/h)
  • miles per hour, (symbol m/h)
  • knots (nautical miles per hour, symbol kt)
  • Mach, where Mach 1 is the speed of sound; Mach n is n times as fast.
Mach 1 ≈ 343 m/s ≈ 1235 km/h ≈ 768 mph (see the speed of sound for more detail)
  • speed of light in vacuum (symbol c) is one of the natural units
c = 299,792,458 m/s
  • the speed of sound in air is about 340 m/s, and 1500 m/s in water
  • Other important conversions
1 m/s = 3.6 km/h
1 mph = 1.609 km/h
1 knot = 1.852 km/h = 0.514 m/s

Vehicles often have a speedometer to measure the speed.

Average speed

Speed as a physical property represents primarily instantaneous speed. In real life we often use average speed (denoted ), which is rate of total distance (or length) and time interval. For example, if you go 60 miles in 2 hours, your average speed during that time is 60/2 = 30 miles per hour, but your instantaneous speed may have varied.

In mathematical notation:

Instantaneous speed defined as a function of time on interval gives average speed:

while instant speed defined as a function of distance (or length) on interval gives average speed:

It is often intuitively expected, but incorrect, that going half a distance with speed and second half with speed , produces total average speed . The correct value is
(Note that the first is a proper arithmetic mean while the second is a proper harmonic mean).

Average speed can be derived also from speed distribution function (either in time or on distance):

Examples of different speeds

Below are some examples of different speeds (see also main article Orders of magnitude (speed)):

  • Speed of a common snail = 0.001 m/s; 0.0036 km/h; 0.0023 mph.
  • A brisk walk = 1.667 m/s; 6 km/h; 3.75 mph.
  • Olympic sprinters (average speed over 100 metres) = 10 m/s; 36 km/h; 22.5 mph.
  • Speed limit on a French autoroute = 36.111 m/s; 130 km/h; 80 mph.
  • Top cruising speed of a Boeing 747-8 = 290.947 m/s; 1047.41 km/h; 650.83 mph.
  • Official air speed record = 980.278 m/s; 3,529 km/h; 2,188 mph.
  • Space shuttle on re-entry = 7,777.778 m/s; 28,000 km/h; 17,500 mph.

Velocity

In physics, velocity is defined as the rate of change of displacement or the rate of displacement. It is a vector physical quantity, both speed and direction are required to define it. In the SI (metric) system, it is measured in meters per second (m/s). The scalar absolute value (magnitude) of velocity is speed. For example, "5 metres per second" is a speed and not a vector, whereas "5 metres per second east" is a vector. The average velocity (v) of an object moving a displacement (s) in a straight line during a time interval (t) is described by the formula:

Simply put, velocity is distance per units of time.

Equations of motion

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

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 to some point in time later .

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

The average velocity of an object undergoing constant acceleration is , where u is the initial velocity and V is the final velocity. To find the displacement, s, of such an accelerating object during a time interval, t, then:

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

also h=-16t2+vt+s

h is the height, v is the velocity, t is the time, and s is the starting height. Usually one will get two answers and must use logic to realize the real answer and what the other one pertains to.

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:

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. In other words only relative velocity can be calculated.

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

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 (also known as velocity made good), and transverse velocity, the component of velocity along a circle centered 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 positive quantities representing counter-clockwise direction and negative quantities representing clockwise direction (in a right-handed coordinate system).

velocity = displacement (divided by) time

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

  • Terminal velocity
  • Hypervelocity
  • Four-velocity (relativistic version of velocity for Minkowski spacetime)
  • Rapidity (a version of velocity additive at relativistic speeds)
  • Orders of magnitude (speed)

Notes


References
ISBN links support NWE through referral fees

  • Halliday, David, Robert Resnick, and Jearl Walker. 2004. Fundamentals of Physics. Wiley; 7 Sub edition. ISBN 0471232319.

External links

Kinematics

← Integrate ... Differentiate →
Displacement (Distance) | Velocity (Speed) | Acceleration | Jerk | Snap

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