People hoping to travel to other worlds are faced with huge challenges.

Question

People hoping to travel to other worlds are faced with huge challenges. One of the biggest is the time required for a journey. The nearest star is 4.1×10^16 m away. Suppose you had a spacecraft that could accelerate at 1.3 g for two thirds of a year, then continue at a constant speed. (This is far beyond what can be achieved with any known technology.)

How long would it take you to reach the nearest star to earth?

Express your answer to two significant figures and include the appropriate units.

t=___________

Answer 1

A simplistic solution follows, with standard kinematics that assume constant acceleration, so we break this into two parts with a₁ = 1.3 g and a₂ = 0. The first duration t₁ = 2/3 (365.24 days) (24 hr/1 day) (3600 s/1 hr) = 2.104 x 10⁷ s [I maintain more than 2 sig figs to avoid round-off error]. The speed at the end of this acceleration, and thereafter, is v₁ = v_o + a₁ t₁ = 0 + 1.3 (9.81 m/s²) (2.104 x 10⁷) = 2.683 x 10⁸ m/s. The displacement during this time is x₁ = 1/2 (v_o + v₁) t₁ = 2.82 x 10¹⁵ m. Total displacement is given as 4.1 x 10¹⁶ m, so displacement during constant velocity phase is Δx = 3.8₂ x 10¹⁶ m. The duration of the latter phase is t₂ = Δx / v₁ = 1.4₂ x 10⁸ s. Total duration of the trip is t₁ + t₂ = 1.6₃ x 10⁸ s • (1 yr/3.156 x 10⁷ s) = 5.2 years. This ignores relativistic effects when traveling at up to 89.5% of c!