Motion on Surface of the Earth: Coriolis Force

A projectile is launched along the x axis at a fixed latitude on the surface of the Earth. The projectile experiences the effect of the Coriolis force, which is a fictitious force observed in a rotating reference frame (blue curve), deflecting the trajectory to the right in the Northern Hemisphere and to the left in the Southern Hemisphere, with respect to an inertial frame (red curve). This effect is important for oceanic currents, prevailing winds, hurricanes, and so on.

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Newton's second law in a rotating coordinate system is

m (OverVector[a])=F-m (OverVector[w])'×(OverVector[r])-2 m (OverVector[w])×(OverVector[v])-m (OverVector[w])×((OverVector[w])×(OverVector[r])),

where the terms on the right-hand side are the transverse force (proportional to the angular velocity), the Coriolis force, and the centripetal force. The equations for the components are

({{m x''==m x (Subscript[ω, 0]^2) (cos^2)(θ)+2 m (Subscript[ω, 0]) y' cos(θ)+m z (Subscript[ω, 0]^2) sin(θ) cos(θ),}, {m y''==-2 m (Subscript[ω, 0]) x' cos(θ)+m y (Subscript[ω, 0]^2) (sin^2)(θ)+m y (Subscript[ω, 0]^2) (cos^2)(θ)-2 m (Subscript[ω, 0]) z' sin(θ),}, {m z''==-m g+m x (Subscript[ω, 0]^2) sin(θ) cos(θ)+2 m (Subscript[ω, 0]) y' sin(θ)+m z (Subscript[ω, 0]^2) (sin^2)(θ).}})

It can be shown that a solution (taking (Subscript[v, (Subscript[y, 0])])=0 and (Subscript[v, (Subscript[z, 0])])=0) is

({{x(t)= (Subscript[v, (Subscript[x, 0])])t-(1/2)  (Subscript[ω, 0]^2) (Subscript[v, (Subscript[x, 0])]) (cos^2)(θ) (t^3),}, {y(t)=(1/3) g  (Subscript[ω, 0]) sin(θ)(t^3)- (Subscript[ω, 0]) (Subscript[v, (Subscript[x, 0])]) cos(θ)(t^2),}, {z(t)=-((g (t^2))/2)-(1/2)  (Subscript[ω, 0]^2) (Subscript[v, (Subscript[x, 0])]) sin(θ) cos(θ)(t^3).}})

Neglecting terms in (Subscript[ω, 0]^2), the solutions are

({{x(t)=(Subscript[v, (Subscript[x, 0])])t,}, {y(t)=(1/3) (Subscript[ω, 0])g sin(θ)(t^3)- (Subscript[ω, 0]) (Subscript[v, (Subscript[x, 0])]) cos(θ)(t^2),}, {z(t)=-((g (t^2))/2),}})

See R. L. Zimmerman and F. I. Olness, Mathematica For Physics, 2nd ed., Reading, MA: Addison–Wesley, 2002.

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Contributed by: Enrique Zeleny

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