Question

The star of a distant solar system explodes as a supernova. At the moment of the explosion, an

resting exploration spaceship is 15 AU away from the shock wave. The shock wave of the explo-

sion travels with 25000 km/s towards the spaceship. To save the crew, the spacecra makes use

of a special booster that uniformly accelerates at 150 m/s2

in the opposite direction.

Determine if the crew manages to escape from the shock wave. (Neglect relativistic eects.)

Answer 1

Let's consider the situation from the frame of reference of the wave. In this frame it does not move and the ship approaches it with the speed of $v = -25000km/s = -2.5\times 10^7m/s$. In order to be in safety, it should achieve a zero speed in this frame. This will happen in time:

$$t = \dfrac{0-v}{a} = \dfrac{-v}{a}$$

where $a = 150m/s^2$ is the acceleration of the ship. In this time the ship covers the following distance (toward the wave):

$$d = vt + \dfrac{at^2}{2} = -\dfrac{v^2}{a}+\dfrac{v^2}{2a} = -\dfrac{v^2}{2a}\\ d = -\dfrac{(2.5\times 10^7)^2}{150}\approx 4.2\times 10^{12}m$$

Since 15 AU is approximately $2.2\times 10^{12}m$ the crew does not manage to escape from the shock wave.

Answer. does not manage.