Propulsive power

If the work to be done is rectilinear motion of a body with constant mass ${\displaystyle m\;}$, whose center of mass is to be accelerated along a (possibly non-straight) line to a speed ${\displaystyle |\mathbf {v} (t)|\;}$ and angle ${\displaystyle \phi \;}$ with respect to the centre and radial of a gravitational field by an onboard powerplant, then the associated kinetic energy is

${\displaystyle E_{K}={\tfrac {1}{2}}m|\mathbf {v} (t)|^{2}}$

where:

${\displaystyle m\;}$ is mass of the body

${\displaystyle |\mathbf {v} (t)|\;}$ 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 + Δt along the path C is defined as the line integral ${\displaystyle \int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{t}^{t+\Delta t}\mathbf {F} \cdot \mathbf {v} (t)dt}$, so the fundamental theorem of calculus has that power is given by ${\displaystyle \mathbf {F} (t)\cdot \mathbf {v} (t)=m\mathbf {a} (t)\cdot \mathbf {v} (t)=\mathbf {\tau } (t)\cdot \mathbf {\omega } (t)}$.

where:

${\displaystyle \mathbf {a} (t)={\frac {d}{dt}}\mathbf {v} (t)\;}$ is acceleration of the center of mass of the body, changing with time.

${\displaystyle \mathbf {F} (t)\;}$ is linear force – or thrust – applied upon the center of mass of the body, changing with time.

${\displaystyle \mathbf {v} (t)\;}$ is velocity of the center of mass of the body, changing with time.

${\displaystyle \mathbf {\tau } (t)\;}$ is torque applied upon the center of mass of the body, changing with time.

${\displaystyle \mathbf {\omega } (t)\;}$ 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 motors do not suffer from this tradeoff, instead trading their high torque for traction at low speed. The power advantage or power-to-weight ratio is then

${\displaystyle {\mbox{P-to-W}}=|\mathbf {a} (t)||\mathbf {v} (t)|\;}$

where:

${\displaystyle |\mathbf {v} (t)|\;}$ is linear speed of the center of mass of the body.

Engine power

The useful power of an engine with shaft power output can be calculated using a dynamometer to measure torque and rotational speed, with maximum power reached when torque multiplied by rotational speed is a maximum. For jet engines the useful power is equal to the flight speed of the aircraft multiplied by the force, known as net thrust, required to make it go at that speed. It is used when calculating propulsive efficiency.

Heat engines and heat pumps

Thermal energy is made up from molecular kinetic energy and latent phase energy. Heat engines are able to convert thermal energy in the form of a temperature gradient between a hot source and a cold sink into other desirable mechanical work. Heat pumps take mechanical work to regenerate thermal energy in a temperature gradient. Standard definitions should be used when interpreting how the propulsive power of a jet or rocket engine is transferred to its vehicle.

Electric motors and electromotive generators

An electric motor uses electrical energy to provide mechanical work, usually through the interaction of a magnetic field and current-carrying conductors. By the interaction of mechanical work on an electrical conductor in a magnetic field, electrical energy can be generated.

Fluid engines and fluid pumps

Fluids (liquid and gas) can be used to transmit and/or store energy using pressure and other fluid properties. Hydraulic (liquid) and pneumatic (gas) engines convert fluid pressure into other desirable mechanical or electrical work. Fluid pumps convert mechanical or electrical work into movement or pressure changes of a fluid, or storage in a pressure vessel.

Thermoelectric generators and electrothermal actuators

A variety of effects can be harnessed to produce thermoelectricity, thermionic emission, pyroelectricity and piezoelectricity. Electrical resistance and ferromagnetism of materials can be harnessed to generate thermoacoustic energy from an electric current.

(Closed cell) batteries

All electrochemical cell batteries deliver a changing voltage as their chemistry changes from "charged" to "discharged". A nominal output voltage and a cutoff voltage are typically specified for a battery by its manufacturer. The output voltage falls to the cutoff voltage when the battery becomes "discharged". The nominal output voltage is always less than the open-circuit voltage produced when the battery is "charged". The temperature of a battery can affect the power it can deliver, where lower temperatures reduce power. Total energy delivered from a single charge cycle is affected by both the battery temperature and the power it delivers. If the temperature lowers or the power demand increases, the total energy delivered at the point of "discharge" is also reduced.

Battery discharge profiles are often described in terms of a factor of battery capacity. For example, a battery with a nominal capacity quoted in ampere-hours (Ah) at a C/10 rated discharge current (derived in amperes) may safely provide a higher discharge current – and therefore higher power-to-weight ratio – but only with a lower energy capacity. Power-to-weight ratio for batteries is therefore less meaningful without reference to corresponding energy-to-weight ratio and cell temperature. This relationship is known as Peukert's law.^[53]

Electrostatic, electrolytic and electrochemical capacitors

Capacitors store electric charge onto two electrodes separated by an electric field semi-insulating (dielectric) medium. Electrostatic capacitors feature planar electrodes onto which electric charge accumulates. Electrolytic capacitors use a liquid electrolyte as one of the electrodes and the electric double layer effect upon the surface of the dielectric-electrolyte boundary to increase the amount of charge stored per unit volume. Electric double-layer capacitors extend both electrodes with a nanoporous material such as activated carbon to significantly increase the surface area upon which electric charge can accumulate, reducing the dielectric medium to nanopores and a very thin high permittivity separator.

While capacitors tend not to be as temperature sensitive as batteries, they are significantly capacity constrained and without the strength of chemical bonds suffer from self-discharge. Power-to-weight ratio of capacitors is usually higher than batteries because charge transport units within the cell are smaller (electrons rather than ions), however energy-to-weight ratio is conversely usually lower.

Fuel cell stacks and flow cell batteries

Fuel cells and flow cells, although perhaps using similar chemistry to batteries, do not contain the energy storage medium or fuel. With a continuous flow of fuel and oxidant, available fuel cells and flow cells continue to convert the energy storage medium into electric energy and waste products. Fuel cells distinctly contain a fixed electrolyte whereas flow cells also require a continuous flow of electrolyte. Flow cells typically have the fuel dissolved in the electrolyte.

Locomotives

A locomotive generally must be heavy in order to develop enough adhesion on the rails to start a train. As the coefficient of friction between steel wheels and rails seldom exceeds 0.25 in most cases, improving a locomotive's power-to-weight ratio is often counterproductive. However, the choice of power transmission system, such as variable-frequency drive versus direct-current drive, may support a higher power-to-weight ratio by better managing propulsion power.

Utility and practical vehicles

Most vehicles are designed to meet passenger comfort and cargo carrying requirements. Vehicle designs trade off power-to-weight ratio to increase comfort, cargo space, fuel economy, emissions control, energy security and endurance. Reduced drag and lower rolling resistance in a vehicle design can facilitate increased cargo space without increase in the (zero cargo) power-to-weight ratio. This increases the role flexibility of the vehicle. Energy security considerations can trade off power (typically decreased) and weight (typically increased), and therefore power-to-weight ratio, for fuel flexibility or drive-train hybridisation. Some utility and practical vehicle variants such as hot hatches and sports-utility vehicles reconfigure power (typically increased) and weight to provide the perception of sports car like performance or for other psychological benefit.

Notable low ratio

This section possibly contains original research. Citations given are mostly primary sources that verify the base statistics but don't assert that the power-to-weight ratio is notable compared to other vehicles. (August 2021)

More information Vehicle, Year ...

Vehicle	Year	Power	Vehicle weight	Power-to-weight ratio
Alberto Contador's Verbier climb 2009 Tour de France on Specialized bike[103]	2009	420 W (0.56 bhp)	62 kg (137 lb)	6.7 W/kg / 0.004 hp/lb
Force Motors Minidor Diesel 499 cc auto rickshaw[107][108]	2008	6.6 kW (8.9 bhp)	700 kg (1,500 lb)	9 W/kg / 0.0054 hp/lb
Mercedes-Benz Citaro O530BZ H2 fuel cell bus[109]	2002	205 kW (275 bhp)	14,500 kg (32,000 lb)	14.1 W/kg / 0.0085 hp/lb
Mazda-Go	1931	10.1 kW (13.5 bhp)	580 kg (1,280 lb)	17.6 W/kg / 0.01 hp/lb
U.S. presidential state car[110][111][112][113][114][115][116]	-	224 kW (300 bhp)	9,000 kg (20,000 lb)	24.7 W/kg / 0.015 hp/lb
International CXT[117]	2004	164 kW (220 bhp)	6,577 kg (14,500 lb)	25 W/kg / 0.015 hp/lb
Amphicar	1965	28.5 kW (38.2 bhp)	1,050 kg (2,310 lb)	27 W/kg / 0.016 hp/lb
Th!nk City[118]	2008	30 kW (40 bhp)	1,038 kg (2,288 lb)	28.9 W/kg / 0.017 hp/lb
Mazda R360	1960	12 kW (16 bhp)	380 kg (840 lb)	31 W/kg / 0.019 hp/lb
Mitsubishi i MiEV[119]	2009	47 kW (63 bhp)	1,080 kg (2,380 lb)	43.5 W/kg / 0.026 hp/lb
Chevrolet Kodiak/GMC Topkick LYE 6.6 L[1][120]	2005	246 kW (330 bhp)	5,126 kg (11,301 lb)	48 W/kg / 0.029 hp/lb
Yamaha PW50 minibike[121][122]	2019	2 kW (2.7 bhp)	41 kg (90 lb)	49 W/kg / 0.03 hp/lb
Mazda Porter (E360)	1975	24 kW (32 hp)	470 kg (1,040 lb)	51 W/kg / 0.03 hp/lb
Honda Beat	1991	47 kW (63 hp)	760 kg (1,680 lb)	62 W/kg / 0.04 hp/lb
Honda Z series "Monkey Bike"[123]	2018	6.9 kW (9.3 bhp)	107 kg (236 lb)	65 W/kg / 0.04 hp/lb
Autozam AZ-1	1992	47 kW (63 hp)	720 kg (1,590 lb)	65 W/kg / 0.04 hp/lb
Subaru 360	1971	27 kW (36 hp)	410 kg (900 lb)	65.5 W/kg / 0.04 hp/lb
Honda N600	1968	34 kW (46 hp)	508 kg (1,120 lb)	66 W/kg / 0.04 hp/lb
Gibbs Aquada[124][125][126][127]	2004	130.5 kW (175.0 hp)	1,750 kg (3,860 lb)	75 W/kg / 0.045 hp/lb

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Sports vehicles

Power-to-weight ratio is an important vehicle characteristic that affects the acceleration of sports vehicles.

This article may contain excessive or inappropriate references to self-published sources. (May 2021)