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)
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.
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