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GeezerVison is showing mathworld.wolfram.com — Acceleration -- from Wolfram MathWorld

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Acceleration

Acceleration is the time derivative of velocity. To obtain formulas for it, let a particle travel a distance s(t) as a function of time t (here, s can be thought of as the arc length of the curve traced out by the particle). The speed (the scalar norm of the vector velocity) is then given by

(ds)/(dt)=sqrt(((dx)/(dt))^2+((dy)/(dt))^2+((dz)/(dt))^2).
(1)

The scalar acceleration is then given by

a=(dv)/(dt)
(2)
=(d^2s)/(dt^2)
(3)
=((dx)/(dt)(d^2x)/(dt^2)+(dy)/(dt)(d^2y)/(dt^2)+(dz)/(dt)(d^2z)/(dt^2))/(sqrt(((dx)/(dt))^2+((dy)/(dt))^2+((dz)/(dt))^2))
(4)
=(dx)/(ds)(d^2x)/(dt^2)+(dy)/(ds)(d^2y)/(dt^2)+(dz)/(ds)(d^2z)/(dt^2)
(5)
=(dr)/(ds)·(d^2r)/(dt^2).
(6)

The vector acceleration is given by

a=(dv)/(dt)
(7)
=(d^2r)/(dt^2)
(8)
=(d^2s)/(dt^2)T^^+kappa((ds)/(dt))^2N^^,
(9)

where T^^ is the unit tangent vector, kappa the curvature, s the arc length, and N^^ the unit normal vector.

Let a particle move along a straight line so that the positions at times t_1, t_2, and t_3 are s_1, s_2, and s_3, respectively. Then the particle is uniformly accelerated with acceleration a iff

a=2[((s_2-s_3)t_1+(s_3-s_1)t_2+(s_1-s_2)t_3)/((t_1-t_2)(t_2-t_3)(t_3-t_1))]
(10)

is a constant (Klamkin 1995, 1996).

Consider the measurement of acceleration in a rotating reference frame. Apply the rotation operator

R^~=(d/(dt))_(body)+omegax
(11)

twice to the radius vector r and suppress the body notation,

a_(space)=R^~^2r
(12)
=(d/(dt)+omegax)^2r
(13)
=(d/(dt)+omegax)((dr)/(dt)+omegaxr)
(14)
=(d^2r)/(dt^2)+d/(dt)(omegaxr)+omegax(dr)/(dt)+omegax(omegaxr)
(15)
=(d^2r)/(dt^2)+omegax(dr)/(dt)+(domega)/(dt)xr+omegax(dr)/(dt)+omegax(omegaxr).
(16)

Grouping terms and using the definitions of the velocity v=dr/dt and angular velocity alpha=domega/dt gives the expression

a_(space)=(d^2r)/(dt^2)+2omegaxv+omegax(omegaxr)+alphaxr.
(17)

Now, we can identify the expression as consisting of three terms

a_(body)=(d^2r)/(dt^2)
(18)
a_(Coriolis)=2omegaxv
(19)
a_(centrifugal)=omegax(omegaxr),
(20)

a "body" acceleration, centrifugal acceleration, and Coriolis acceleration. Using these definitions finally gives

a_(space)=a_(body)+a_(Coriolis)+a_(centrifugal)+alphaxr,
(21)

where the fourth term will vanish in a uniformly rotating frame of reference (i.e., alpha=0). The centrifugal acceleration is familiar to riders of merry-go-rounds, and the Coriolis acceleration is responsible for the motions of hurricanes on Earth and necessitates large trajectory corrections for intercontinental ballistic missiles.

See also

Angular Acceleration, Arc Length, Convergence Improvement, Jerk, Velocity

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  • centripetal acceleration formula
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References

Klamkin, M. S. "Problem 1481." Math. Mag. 68, 307, 1995.Klamkin, M. S. "A Characteristic of Constant Acceleration." Solution to Problem 1481. Math. Mag. 69, 308, 1996.

Referenced on Wolfram|Alpha

Acceleration

Cite this as:

Weisstein, Eric W. "Acceleration." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Acceleration.html

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