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Online Calculator: Journey Energy & Power

{CalQlata © 01/01/20}

This calculator calculates the amount of energy required to complete a journey, along with the power needed to accomplish each stage.
It segregates the journey into [up to] ten altitude variations (10-stages).

Energy may be used to determine the quantity of fuel required to complete the journey, or any stage thereof. Fuels are generally qualified by the energy available per unit mass (Joules per kilogram or foot-pounds per pound).
Power is the rate of energy expenditure (energy intensity) required to complete a given stage of the journey. This value may be used to calculate the rate of fuel consumption (kilometres per litre or miles per gallon) for that particular stage.
This calculator takes into account rolling resistance and wind resistance.


journey dimensions

Common Data:

m = travelling mass.
A = cross-sectional area of the travelling mass.
ρ = air density through which the mass is travelling.
vₘₒₙ = nominal velocity used in the equivalent energy calculation(km÷1.609344).
Cd = drag coefficient of the travelling mass.
μ = frictional resistance in all of the mechanical components resisting forward movement.

m A ρ vₙₒₘ Cd μ
kg m² kg/m³ mph

Input Data:

Each of the following applies [only] to the associated stage:
distance = horizontal (or map) distance travelled.
rise = difference in altitude. If the surface is flat, a zero (0) must be entered.
velocity = the actual constant speed of travel over the surface; not the horizontal speed (unless the surface is flat).

Delete the unwanted value (Imperial or metric) before calculating. If values exist in both Imperial and metric cells, this calculator will default to the Imperial value.
Irrespective of the entered units, this calculator will convert the values to metres, kilograms and seconds for the calculation results, which will be provided in metric (Joules and Watts).

stage distance rise velocity
yards metres feet metres mph kph
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Output Data:

Each of the following applies [only] to the associated stage:
Eₐ = energy required to overcome resistance to atmospheric gases.
Eₕ = horizontal energy in the mass.
Eᵥ = vertical energy in the mass.
E = total energy expended in the mass.
P = power required to achieve the velocity entered.

stage Ea Eₕ Eᵥ E P
J J J J W
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Flat surface Equivalent Velocity:

In order to consume the same energy (or fuel quantity) as for the calculated journey, whilst travelling over a [relatively] flat surface at a nominal velocity (vₙₒₘ), you would need to cover the following distance (and time):
dₑ = metres ( calculated (above) total: m)
tₑ = seconds ( calculated (above) total: s)

Intermediate Output Data:

Each of the following applies [only] to the associated stage:
t = time taken.
θ = angle of the slope [°].
d = actual distance (along the surface of the slope) of travel.
vₕ = horizontal velocity of travel.
vᵥ = vertical velocity of travel.
aᵥ = vertical acceleration (excluding gravitational).

These results are provided for information only. They were used in the above energy and power calculations.

stage t θ d vₕ vᵥ aᵥ
s ° m m/s m/s m/s²
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Fuel Consumption

The expected quantity of fuel needed to complete your journey is:
Fuel specific internal energy; J/kg
Fuel specific gravity; (fuel density ÷ water density)
Eₜ = Joules, which needs;
litres, or
UK gallons, or
US gallons
assuming your drive mechanism is 100% efficient!


Help

This Journey calculator is accessible from anywhere in the website using the shortcut key; "Alt" + "j".
The "Reset" button clears all calculations on the page and reinstalls default values (this button may not respond in the FireFox browser).
Reset can also be achieved by pressing the "F5" key.
Hover your cursor over the symbols for an associated description.

Calculated Energies

The component energies (see above image), that together constitute the total; E.

Horizontal
Eₕ₁; kinetic energy due to horizontal velocity (vₕ)
Eₕ₃; frictional resistance (μ) due to vertical acceleration (aᵥ = vᵥ²/dᵥ) excluding gravitational

Vertical
Eᵥ₁; kinetic energy due to vertical velocity (vᵥ)
Eᵥ₂; potential energy due to vertical acceleration (aᵥ = vᵥ²/dᵥ) excluding gravitational

Incline
E₁; resolved vertical and horizontal energies √[Eₕ² + Eᵥ²]
E₂; air resistance energy
E₃; potential energy due to gravitational acceleration (aᵥ = g)

Total
E; E₁+E₂+E₃ used to calculate fuel consumption.

Rolling Resistance

Rolling resistance refers to the resistance to movement of mass (m) excluding the effect of atmospheric drag and added mass. It does not only apply to the wheels of a vehicle. A pedestrian will also experience resistance to movement through the frictional resistance in the pedestrian's footwear.
This force is induced by the vertical acceleration in the mass due to the power expended by equivalent vertical acceleration due to vᵥ (vᵥ²/dᵥ).
The value entered for 'μ' is the same thing as frictional coefficient. It is always established by experimentation and therefore a guestimate. For the purposes of this calculator, you may assume the following approximations unless more accurate information is available:
car; 0.12
lorry; 0.16
train; 0.1
pedestrian; 0.07

Wind Resistance

This calculation is based upon the mass travelling through a stationary atmosphere. In other words, 'wind' resistance on the travelling mass comes only from its own velocity.
Resistance to drag is calculated using Morison's formula and based upon the coefficient you enter above (Cd). You may assume that most modern cars have a value of ≈0.4, lorries will be closer to ≈0.65 and a pedestrian will be 0.7.

Energy

Energy is a force multiplied by a distance. Its [metric] units are kg x m/s² x m (mass x acceleration x distance). This calculation tells us how much of it is required to complete the journey (or any stage thereof), which enables us to determine the quantity of fuel required.
Fuel energy capacity is usually specified per unit mass (J/kg), which gives us the means to calculate the quantity required; bearing in mind that no engine is 100% efficient.
An interesting calculation would be to map out a journey, enter the details above, calculate the expected fuel consumption for the journey. Then follow the journey in your car (at the velocities entered) and see how much fuel you actually use. Expected÷Actual will give you your car's efficiency.

Power

Power is energy intensity, in other words, the rate of energy expenditure (per unit time).
The more power available, the faster (velocity) your vehicle could travel over that stage of your journey.
Moreover, the harder a mass accelerates the more power it needs;
P = E/t = m.a.d/t = F.v {kg . m/s² . m / s = kg . m/s² . m/s}

Equivalent Energy Calculation

It is assumed that the calculated journey comprises a number of stages each of which slopes up or down-hill.
This calculation tₑ, dₑ provides an equivalent journey over a flat surface based upon a nominal velocity; vₙₒₘ
Whilst this calculation is provided for information only, it may be useful in determining, for example, an equivalent [walking] exercise distance to your normal exercise over your own hilly terrain.

Fuel

To determine the quantity (mass) of fuel required for a journey, you need to divide the above calculated energy (E) by the specific internal energy (u) of the preferred fuel; fuel mass = E/u
The units of energy calculated above are in Watts. If you are calculating for a pedestrian, you may prefer the energy in units of calories (food), the conversion for which is;
1 Joule = 0.2386623 calories.

Fuel Consumption

Fuel consumption is defined simply by the fuel mass per unit time {kg/s}. For example, say we assume the fuel to be petrol (the default value above), for no other reason than it is currently the cleanest and most efficient.
The specific internal energy of Petrol is ≈48E+06 J/kg
and its density is approximately 800 kg/m³ (S.G. = 0.8)
Liquid volume is 1000 litres per m³ (1 litre = 0.22 UK gallons or 0.264 US gallons).
Therefore;
1 litre of petrol holds 4.8E+07 x 0.8; Eₑ = 3.84E+07 Joules of energy
1 UK gallon of petrol holds 4.8E+07 x 0.8 x 4.54609; Eₑ = 1.746E+08 Joules of energy
1 US gallon of petrol holds 4.8E+07 x 0.8 x 4.54609; Eₑ = 1.454E+08 Joules of energy
You can work out your own fuel consumption simply by dividing the above total energy by the relevant energy capacity;
Q = Eₑ/Eₜ (litres or gallons).
The above calculation includes rolling (or frictional) resistance (μ),

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