Calculate Motor Size

July 2004

Based on the basic formulas for the physics of motion we can calculate the theoretical energy needs to move a vehicle.

There are three major forces at work which resist a vehicle from moving:

1. Rolling resistance
2. Air resistance
3. The force of gravity as a vehicle moves up a hill

Rolling Resistance

The force of the rolling resistance is a function of the weight of the vehicle multiplied by a coefficient of the rolling resistance.

This force is mostly independant of car speed.

Froll = uroll M agravity

Where:
    u_roll - is the rolling resistance coefficient. Typical values for the rolling resitance coeffiecient (u_roll) range from 0.010 to 0.015
    M - is the mass of the vehicle (kg)
    a_gravity - is the force of gravity (9.8 m/s²)

Air Resistance

The force of the air resistance is proportional to the square of the speed, the density of the air, the silhoette area of the car, and the drag coeffecient for the vehicle.

Fair = 1/2 Cd A p v2

Where:
    C_d - is the Drag coefficient (no dimension)
    A - is the surface area of the car (m²)
    p - is the density of the air (1.2 kg/m³ at sea level at normal temperatures)
    v - is the speed of the vehicle (m/s)

Here is a table of common drag coefficients for some vehicles:

Vehicle	Cd
VW New Beetle	0.38
Porsche Carrera	0.38

Force required to go uphill

The force required to lift a car uphill is a function of the angle of the hill and the force of gravity

Fhill = (% grade of hill) M agravity

Where:
    M - is the mass of the vehicle (kg)
    a_gravity - is the force of gravity (9.8 m/s²)

Power required for Forward Motion

Power is a measurement of work per unit time. Work is a measurement of a force moved some distance. Therefore to determine the power required to move a vehicle at a certain speed, it is simply the total of all forces to overcome multiplied by the speed.

P = (Froll + Fair + Fhill) v

Power required for Acceleration

To calculate the power required to accelerate a vehicle, first determine the amount of energy required to accelerate a vehicle from 0 to a speed v:

Ek = 1/2 M v2

Where:
    M - is the mass of the vehicle (kg)
    v - is the final speed of the vehicle

then to calculate the power, divide the energy required by the time it takes to accelerate the vehicle:

P = Ek/t

Where:
    E_k - is the energy required to accelerate the vehicle to the speed
    t - is the time is takes to accelerate the vehicle to the speed

Putting it all together

References

• Automotive Power a paper of how to calculate these forces.