Can you jump a motorcycle into space?

[This question and answer are intended as an example of part of a tutorial on the conservation of energy for an A-Level physics student.]

Question:

A daredevil is planning the ultimate feat: jumping a motorcycle into space from a standstill. How big does her fuel tank need to be?

Assume the following:

Space begins at 100 km above sea level.

The acceleration due to gravity is 9.81 ms^-2 regardless of altitude.

The energy density of fuel is 50 MJ/kg.

The motorcycle and rider together weigh 500 kg.

No energy is lost by the rider and motorcycle to the surroundings.

Answer:

The simplest way to tackle this problem is by conservation of energy. As with most problems of this type, we’ll look separately at the initial situation and the final situation, then assume no energy is lost in between.

Initial situation:

The motorcycle, rider and fuel are at sea level. Call the mass of the motorcycle and rider together m, and the mass of the fuel M. The only energy in the system is the chemical energy U of the fuel:

E_initial = U,

and U = uV,

where u is the energy density of the fuel and V is its volume.

Final situation:

If the rider only just reaches her goal, then she and the motorcycle only just reach the 100 km level before falling back to Earth. At that point, there is no fuel left, and they are still for just an instant. Therefore the only energy in the system is then the gravitational potential energy of the rider and motorcycle:

E_final = mgh,

where h = 100 km.

Conservation of energy:

Because no energy was lost to the surroundings, we have E_final = E_initial, so:

U = mgh,

or

uV = mgh.

We want the volume of the fuel tank, so:

V = mgh/u.

Substituting in all the values, we obtain:

V = (500 kg)(9.81 ms^-2)(100 km) / (50 MJ/kg)

Which evaluates to:

V = 9.8 m³ = 9,800 L.