K5    The "Conquest of Space"

It is a nice exercise in mathematics and physics to calculate a human voyage to a nearby star. Human, here, means that during the acceleration at the start and during the deceleration before the return the passengers should feel a constant acceleration equivalent to the gravitational force felt on the surface of the earth. Given the desired cruise velocity after the acceleration phase and the distance to the destination then one has all of the information needed.

Kranzer investigates in [43] such a trip to the nearest star α Centauri (distance about 4.2 light years) at a speed of 0.9 • c after the acceleration phase. He uses a few formulas without showing their derivation. For interested students this is a nice challenge! Calculus at the high school level is sufficient to do the calculations. The starting point is the following equation (see also section E4!)

After division by m0 one obtains for the speed of the space vehicle, the following differential equation

Students can at least verify that the following functions for v(t) and x(t) satisfy this differential equation:

The space traveler’s elapsed proper time during the acceleration phase is then calculated by taking into account that the following always applies

Forming the integral on the right side for the duration of the acceleration phase in coordinate time t for an observer on the earth, yields the elapsed proper time Δτ for a passenger in the spaceship. The result is (for v0 = 0 and x0 = 0)

The journey there and back takes around 12 years for the earth-bound people, while more than three and a half years elapse on-board. How one can technically achieve a cruise velocity of 0.9 • c for three-quarters the start mass (!)  To speak of an impending "conquest of space" is extremely exaggerated.

For people who like to do calculations, [27] contains a lot of stimulating material on pages 164 to 230!  Also [25-161ff] discusses what the STR has to say about travel in our cosmic neighborhood.