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In classical mechanics, areal velocity (also called sector velocity or sectorial velocity) is a pseudovector whose length equals the rate of change at which area is swept out by a particle as it moves along a curve. It has SI units of square meters per second (m 2 /s) and dimension of square length per time L 2 T −1.
Since the speed v is likewise unchanging, the areal velocity 1 ⁄ 2 vr ⊥ is a constant of motion; the particle sweeps out equal areas in equal times. The area A of a circular sector equals 1 ⁄ 2 r 2 φ = 1 ⁄ 2 r 2 ωt = 1 ⁄ 2 r v φ t. Hence, the areal velocity dA/dt equals 1 ⁄ 2 r v φ = 1 ⁄ 2 h.
A rocket's required mass ratio as a function of effective exhaust velocity ratio. The classical rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity and can thereby move due to the ...
The Frenet–Serret formulas apply to curves which are non-degenerate, which roughly means that they have nonzero curvature. More formally, in this situation the velocity vector r′(t) and the acceleration vector r′′(t) are required not to be proportional. Let s(t) represent the arc length which the particle has moved along the curve in ...
This constant areal velocity can be calculated as follows. At the apapsis and periapsis, the positions of closest and furthest distance from the attracting center, the velocity and radius vectors are perpendicular; therefore, the angular momentum L 1 per mass m of the particle (written as h 1) can be related to the rate of sweeping out areas
The term 1 / ρ 2 ∇ρ × ∇p is the baroclinic term. It accounts for the changes in the vorticity due to the intersection of density and pressure surfaces. The term ∇ × ( ∇ ∙ τ / ρ ), accounts for the diffusion of vorticity due to the viscous effects. The term ∇ × B provides for changes due to external body forces.
The speed in the formula is squared, so twice the speed needs four times the force, at a given radius. This force is also sometimes written in terms of the angular velocity ω of the object about the center of the circle, related to the tangential velocity by the formula v = ω r {\displaystyle v=\omega r} so that F c = m r ω 2 ...
This is the formula for the relativistic doppler shift where the difference in velocity between the emitter and observer is not on the x-axis. There are two special cases of this equation. The first is the case where the velocity between the emitter and observer is along the x-axis. In that case θ = 0, and cos θ = 1, which gives: