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How much time would be required for a starship to reach Proxima Centauri, the star closest to the Sun, at a distance of L=4.244L=4.244 light years(ly)? Assume that it starts from rest, maintains a comfortable acceleration magnitude of 1.000 g1.000 \mathrm{~g} for the first 0.0450 y0.0450 \mathrm{~y} and a deceleration (slowing) magnitude of 1.000 g1.000 \mathrm{~g} for the last 0.0450 y0.0450 \mathrm{~y}, and cruises at constant speed in between those periods.

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Givens

  • L=4.244 ly=41016 mL=4.244\ \text{ly}=4\cdot 10^{16}\ \text{m} - the distance to Proxima Centauri
  • a1=a3=1 g=9.81 ms2a_1=a_3=1\ g=9.81\ \frac{\text{m}}{\text{s}^2} - the accelerations in the first and the third part
  • Δt1=Δt3=0.045 y=1.42106 s\Delta t_1=\Delta t_3=0.045\ \text{y}=1.42\cdot 10^6\ \text{s} - the duration of the first and the third part

Required

  • τ\tau - the time it takes to reach Proxima Centauri

How can we use the known expressions for the speed and the distance travelled by the ship in order to find τ\tau?

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