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 explosion travels with 25000 km/s towards the spaceship. To save the crew, the spacecraft 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 effects.)

Answer 1

Let us determine the equations of motion of the shock and of the spaceship. Let the supernova be in the origin of the coordinate system. Therefore, the x-coordinate of the shock wave will be

$x_1(t) = v_{\text{shock}}\cdot t= 2.5\cdot10^7\,\mathrm{m/s}\cdot t.$

The spaceship is initially at the point with the coordinate $x_0 =15 \,\text{AU} = 15\cdot 1.5\cdot10^{8}\,\mathrm{km} = 2.25\cdot10^{12}\,\mathrm{m}.$

The spaceship moves with constant acceleration, so its coordinate after t seconds will be

$x_2(t) = x_0 + v_0t + \dfrac{at^2}{2} = 2.25\cdot10^{12}\,\mathrm{m} + 0 + \dfrac{150\cdot t^2}{2}.$

The crew will escape if its coordinate is greater than the coordinate of shock. Let us determine if the shock overtakes the spaceship

$2.25\cdot10^{12} + \dfrac{150\cdot t^2}{2} = 2.5\cdot10^7\cdot t,\\ 4.5\cdot10^{12} + 150\cdot t^2 = 5\cdot10^7\cdot t,\\ 150\cdot t^2 - 5\cdot10^7\cdot t + 4.5\cdot10^{12} = 0.$

$D = b^2 - 4ac = (5\cdot10^7)^2 - 4\cdot150\cdot4.5\cdot10^{12} = -2\cdot10^{14}< 0,$ so there are no real roots of the equation. That means, the spaceship will always be ahead the shock.