Question

The star of a distant solar system explodes as a supernova .At the moment of the explosion ,an resting exploration spaceship is AU away from the shock wave . The shock wave of the explosion spaceship travels with 25000km/s towards the spaceship .To save the crew , the spacecraft makes use of a special booster that uniformly accelerates at 150m/s^2 in the opposite direction .

Determine if the crew manages to escape from the shock wave (neglect relativistic effects)

Answer 1

Let the x-axis be directed from star to the starship. At the initial moment the coordinate of the ship is $1\,\text{AU}= 150\text{ mln\, km}= 1.5\cdot10^{11}\,\mathrm{m}$ and this is the initial distance between the shock and the ship. After the beginning of the acceleration the coordinate of the ship after time t will be $x=1.5\cdot10^{11}\,\mathrm{m} + a t^2/2 = 1.5\cdot10^{11}\,\mathrm{m} + 75\,\mathrm{m/s^2}\cdot t^2.$

The coordinate of the shock wave is $x_w = 25\cdot10^6\,\mathrm{m/s}\cdot t$ . Let us determine if there is a moment t at which the shock wave reaches the starship: $x=x_w.$

$1.5\cdot10^{11}\,\mathrm{m} + 75\,\mathrm{m/s^2}\cdot t^2 - 25\cdot10^6\,\mathrm{m/s}\cdot t = 0.$

We calculate the discriminant: $D=(25\cdot10^6)^2 - 4\cdot75\cdot 1.5\cdot10^{11} = 5.8\cdot10^{14} > 0.$

The roots of the equation are

$t_{1,2} = \dfrac{25\cdot10^6 \pm \sqrt{5.8\cdot10^{14}}}{2\cdot75}, \;\; t_1 =6.1\cdot10^3\,\mathrm{s} ,\; t_2= 3.3\cdot10^5\,\mathrm{s}.$

Therefore, after approximately 6000 seconds the shock wave will reach the ship.