To calculate the time it would take to travel to Proxima Centauri (4.25 light-years away) in a ship under constant acceleration/deceleration at 1G (9.81 m/s²), we can use the equations of motion from special relativity.
From the perspective of an observer on Earth, we need to consider the time experienced by the traveler on the ship due to time dilation effects. However, for simplicity, we'll assume the ship accelerates instantaneously to 1G and maintains a constant velocity until it reaches the midpoint, then decelerates at 1G until it reaches Proxima Centauri.
Let's calculate the time it takes from the perspective of the ship first:
- Acceleration phase: The ship accelerates at 1G until reaching the midpoint (halfway to Proxima Centauri). The equation to calculate the time taken during this phase is given by:
t_acc = sqrt((2 * d) / a),
where d is the distance to the midpoint (2.125 light-years or 2.0145 × 10^16 meters) and a is the acceleration (9.81 m/s²).
- Deceleration phase: The ship decelerates at 1G from the midpoint until reaching Proxima Centauri. The time taken during this phase is the same as the acceleration phase:
t_dec = t_acc.
- Total ship time: The total ship time is the sum of the acceleration and deceleration times:
t_ship = t_acc + t_dec.
Now, let's calculate the time as seen from Earth, accounting for time dilation:
t_earth = t_ship * sqrt(1 - (v^2 / c^2)),
where v is the velocity of the ship and c is the speed of light.
Since the ship is constantly accelerating at 1G, the velocity increases linearly with time. At the midpoint, the ship's velocity would be half of the maximum velocity reached at the end. Therefore, the maximum velocity (v_max) can be calculated as:
v_max = a * t_acc.
Now we can substitute the values and calculate the times:
t_acc = sqrt((2 * 2.0145 × 10^16) / 9.81) ≈ 20.41 years, t_dec = t_acc ≈ 20.41 years, t_ship = t_acc + t_dec ≈ 40.82 years.
To calculate the velocity and time dilation at the midpoint, we need to know the relation between velocity and time during acceleration. Using the equation:
v = a * t,
we can solve for t when v = v_max / 2:
t = (v_max / 2) / a = (a * t_acc / 2) / a = t_acc / 2 ≈ 10.205 years.
Now, we can calculate the velocity at the midpoint:
v_midpoint = a * t = 9.81 * 10.205 ≈ 100.16 m/s.
Finally, we can calculate the time dilation factor:
time_dilation_factor = sqrt(1 - (v_midpoint^2 / c^2)) ≈ 0.99999999999941.
Now we can calculate the time as seen from Earth:
t_earth = t_ship * time_dilation_factor ≈ 40.82 * 0.99999999999941 ≈ 40.82 years.
So, from the perspective of the ship, it would take approximately 40.82 years to reach Proxima Centauri. From the perspective of Earth, accounting for time dilation effects, it would also take approximately 40.82 years.