Editing Power-to-weight ratio (section)

====Propulsive power====
If the work to be done is [[wikt:rectilinear|rectilinear]] motion of a body with constant [[mass]] <math>m\;</math>, whose [[center of mass]] is to be accelerated along a (possibly non-straight) line to a speed <math>|\mathbf{v}(t)|\;</math> and angle <math>\phi\;</math> with respect to the centre and [[Spherical coordinate system|radial]] of a [[gravitational field]] by an onboard [[powertrain|powerplant]], then the associated [[kinetic energy]] is

:<math> E_K =\tfrac{1}{2} m|\mathbf{v}(t)|^2 </math>

where:
:<math>m\;</math> is mass of the body
:<math>|\mathbf{v}(t)|\;</math> is speed of the [[center of mass]] of the body, changing with time.

The [[work–energy principle]] states that the work done to the object over a period of time is equal to the difference in its total energy over that period of time, so the rate at which work is done is equal to the rate of change of the kinetic energy (in the absence of potential energy changes).

The work done from time ''t'' to time ''t'' + Δ''t'' along the path ''C'' is defined as the [[line integral]] <math>\int_C \mathbf{F} \cdot d\mathbf{x} = \int_t^{t + \Delta t} \mathbf{F} \cdot \mathbf{v}(t) dt</math>, so the [[fundamental theorem of calculus]] has that power is given by <math>\mathbf{F}(t) \cdot \mathbf{v}(t) = m\mathbf{a}(t) \cdot \mathbf{v}(t) = \mathbf{\tau}(t) \cdot \mathbf{\omega}(t)</math>.

where:
:<math>\mathbf{a}(t) = \frac{d}{dt}\mathbf{v}(t)\;</math> is acceleration of the [[center of mass]] of the body, changing with time.
:<math>\mathbf{F}(t)\;</math> is linear force&nbsp;– or thrust&nbsp;– applied upon the center of mass of the body, changing with time.
:<math>\mathbf{v}(t)\;</math> is [[velocity]] of the center of mass of the body, changing with time.
:<math>\mathbf{\tau}(t)\;</math> is [[torque]] applied upon the center of mass of the body, changing with time.
:<math>\mathbf{\omega}(t)\;</math> is [[angular velocity]] of the center of mass of the body, changing with time.

In [[propulsion]], power is only delivered if the powerplant is in motion, and is transmitted to cause the body to be in motion. It is typically assumed here that mechanical transmission allows the powerplant to operate at peak output power. This assumption allows engine tuning to trade [[power band]] width and engine mass for transmission complexity and mass. [[Electric motor]]s do not suffer from this tradeoff, instead trading their high [[torque]] for [[Traction (engineering)|traction]] at low speed. The [[Mechanical advantage|power advantage]] or power-to-weight ratio is then

:<math> \mbox{P-to-W} = |\mathbf{a}(t)||\mathbf{v}(t)|\;</math>

where:
:<math>|\mathbf{v}(t)|\;</math> is linear speed of the [[center of mass]] of the body.