Energy required for flying cars

Lets calculate how much energy the air propelled flying car consumes.


Near the surface of Earth, the force of gravity is F_g = m_car * g, where m_car is the mass of the car and g is the gravitational acceleration.

To hover in the air, the car must produce a force equal in magnitude and opposite in direction by propelling the mass of air downwards. The mass of air will have to be accelerated from velocity 0 to v per time t, F_lift = m_air * a_air.

Both forces, gravitational and lift must be equal for a car to hover, F_g = F_lift, gives us:
(m_car/m_air) = a_air / g !!!!!!!
This means if the mass of accelerated air is equal to the mass of a car, then the air must be accelerated from 0 to 9.8 m/s in one second. If we mass of accelerated air is 10 times larger, then a_air must be then time smaller than g and equal to 0.98 m/s in one second.


The energy required to accelerate this mass of air is E_kinetic = 1/2 * m_air * v_air^2. This energy is a theoretical minimum energy required for a car to hover in the air. Less energy is required to move larger mass of air at lower velocities, because of v^2.

Example 1. 1000 kg car:
a) m_air = 1000 kg of air accelerated downwards to speeds 9.8 m/s each second requires energy E = 0.5*1000 kg * 9.8 (m/s)^2 = 48 kJ each second = 48 kW.
b) m_air = 10000 kg of air accelerated downwards to speeds 0.98 m/s each second requires energy E = 0.5 * 100 kg * 0.98 (m/s)^2 = 480 J each second = 480 W. That is 10000 m3 of air passing the propeller blades per second!

The real question about flying cars is: What is maximum mass of air we can accelerate with the given (limited) dimensions of the car?



Example 2.
It would impractical to have private flying cars larger wider and longer than ~ 10 m x 10 m. So the area of propeller blades is 100 m2.
Typical winds are 10 m/s (20 knots), typical strong winds are 100 m/s.

a) v_air = 100 m/s, the total mass of air that will be accelerated per second is 100 m/s * 100 m2 = 10000 m3 /s =~ 10000 kg / s. Mass of the car is: m_car/m_air = v_air/g = 100,000 kg. Energy required is: E = 0.5*10000kg*(100m/s)^2 = 50 kJ per s = 50 MJ per s = 50 MW.

b) v_air = 10 m/s, the total mass of air that will be accelerated per second is 10 m/s * 100 m2 = 1000 m3 /s =~ 1000 kg / s. Mass of the car is: m_car/m_air = v_air/g = 1000 kg. Energy required is: E = 0.5*1000kg*(10m/s)^2 = 50 kJ per s = 50 kW.
Its not that much! Typical small car engines are ~50 kW. Tesla car batteries are ~ 50 kWh, which means that the hovercraft can travel for 1 hour at the energy cost of $0.25 / kWh * 50 kWh = $12.5.
Since there not traffic jams in the air, the hovercraft does not need to stop moving and waste energy for hovering.

Graphs below shows the maximum mass allowed and the power consumption of hovering aircraft as a function of air velocity and blade area.