Speed & Velocity

Summary

• Speed is the rate of change of distance with time.
  • As a scalar it has magnitude only.
  • Average speed…
    • is measured over a non-zero time interval and
    • is represented by the symbol v_ave or v̅ (overline)
  • Instantaneous speed…
    • is the limit of average speed as the time interval approaches zero,
    • is the first derivative of distance with respect to time, and
    • is represented by the symbol v (italic)
• Velocity is the rate of change of displacement with time.
  • As a vector it must be stated with both magnitude and direction.
  • Average velocity…
    • is measured over a non-zero time interval and
    • is represented by the symbol v_ave or v̅ (overline)
  • Instantaneous velocity…
    • is the limit of average velocity as the time interval approaches zero,
    • is the first derivative of displacement with respect to time, and
    • is represented by the symbol v (boldface)
• The various forms of speed and velocity are defined by the following equations…

  average speed		v̅ =  Δs Δt
  instantaneous speed		v =  lim Δs  =  ds Δt → 0 Δt dt
  average velocity		v̅ =  Δr Δt
  instantaneous velocity		v =  lim Δr  =  dr Δt → 0 Δt dt

• The relation of speed to velocity
  • An object's average speed approaches the magnitude of its average velocity as the time interval approaches zero.

    Δt → 0

    ⇒

    v̅ → |v̅|

  • The instantaneous speed of an object is the magnitude of its instantaneous velocity.

    v = |v|

• The SI unit of speed and velocity is the meter per second [m/s].