Deceleration times

Notation

Spacecraft initial speed: \(v_0 = 6 \times 10^6\)

Spacecraft final speed (hoverball maximum speed): \(v_1\)

Deceleration start time: \(t_0 = 0\)

Deceleration end time: \(t_1\)

Uniform deceleration

Space craft decelerates at a constant rate, \(a = \frac{v_1 - v_0}{t_1}\).

(NB \(a\) will be a negative number.)

Velocity given by: \(v = v_0 + at\)

Because \(v_1\) is so much less than \(v_0\) it’s difficult to see, from the graph above that all the lines end at different velocities.

This can more easily be seen in the expanded graph below.

Uniform decrease in kinetic energy

This model assumes that the decrease in kinetic energy is uniform, leading to a much more rapid slowing towards the end. This experience is familiar to car drivers and the model underlies the braking distances quoted in the UK Highway Code.

The formula is:

\(v = \sqrt{v_0^2 + kt}\ \),

where \(k = \frac{v_1^2 - v_0^2}{t_1}.\)

Exponential decrease in velocity

This model assumes the speed falls rapidly at first and then more slowly.

The formula is:

\(v = ae^{-bt}\ \),

where \(a = v_0\ \) and

\(b = \frac{1}{t_1}\ln{\frac{v_0}{v_1}} \ \)