Travel to nearest star

Classical Mechanics Level 1

An interstellar trip is being planed in a Sci-fi movie. The trip will start from Earth and end at Proxima Centauri. The distance traveled will be approximately 4.25 light years. The space craft has the capability of maintaining a constant rate of acceleration or deacceleration over the entire trip. For the long-term heath and comfort of the crew the maximum acceptable acceleration rate will be strictly limited to 10 m/s^2. The trip will end as the space craft arrives in a low stable orbit around an exoplanet near Proxima Centauri.

What is the minimum time required for the space craft to complete the trip? Give your answer in days rounded down, i.e. floor value of the number of days required for the trip.

Details and assumptions

In the fictional movie universe. Newton’s laws remain valid; i.e. F= ma will hold true. Relativity theory does not apply. i.e. There are no relativity theory based relationships concerning mass, time, energy or velocities.

Maximum acceleration at any time throughout the trip = $10\frac { m }{ { s }^{ 2 } }$

Total distance traveled = 4.0 E+13 km The distance between the earth and the exoplanet does not change.

Initially the ship is orbiting Earth and ends the trip orbiting an exoplanet near Proxima Centauri. The initial and final velocities of the spaceship can be considered as negligible.

The answer is 1464.

1 solution

Steven Chase
Dec 3, 2020

import math

# assume that the craft accelerates at 10 m/s^2 to the half-way point...
# ......before beginning to decelerate at the same rate
# calculate the time to the half-way point and double it

a = 10.0                     # acceleration
Dhalf = 2.0 * (10.0**(16.0))     # distance to half-way point in meters

# Dhalf = 0.5*a*(t**2.0)         # kinematic equation to solve for time

tsq = 2.0*Dhalf/a
t = math.sqrt(tsq)           # time to half-way point in seconds

t = 2.0*t                     # time for full trip

day = 24.0*60.0*60.0          # number of seconds in a day

print (t/day)                 # number of days for trip
#1464.01743526

So almost exactly four years? Sounds great until you remember it's a $4.25$ light-year distance...(or is that what "Relativity theory does not apply" means?)

Chris Lewis - 6 months, 1 week ago

Yes! in this problem the ship maximum speed will be much higher then the speed of light. In fact, as you have pointed out, the ships average speed is greater then the speed of light. This is not possible when considering the implications of the theory of relativity. To avoid very complex physics and math this movie universe is ignoring relativity theory. as is often the case in fiction , the space craft's velocity is not limited and can travel faster then the speed of light.

With my very limited and overly simplified understanding of relativity theory I believe there would be a number of factors that would greatly complicate the problem if relativity theory was applied. The velocity of the spacecraft would increase gradually towards the speed of light relative to the earth and Proxima Centauri. I believe that relativity theory would indicate that the the mass of the ship would increase exponentially. As the ships speed approached the speed of light the mass of the ship would approach infinity. An infinity massive object could not be accelerated without a infinite amount of force. a=f/m. an infinite force acting through a distance would require an infinite amount of energy. These effects would not happen suddenly, but build up exponentially as the ship velocity increased. the problem solution would be much more involved when relativity theory is included.

Darryl Dennis - 6 months, 1 week ago

It's a weird coincidence that accelerating at the gravitational acceleration we happen to be used to over the distance to our (second) nearest star would average out at so close to the speed of light.

I'm not sure what the weirdest/most noticeable implications of a universe in which faster than light travel was possible would be (I guess better to say a universe with no general relativity for this case).

The assumption that is casually mentioned a lot in mechanics problems that I think would have the craziest results is "assuming there's no friction...".

Chris Lewis - 6 months, 1 week ago

One of the "details and assumptions" is that relativity doesn't apply. That was one of my first thoughts upon seeing this problem as well.

Steven Chase - 6 months, 1 week ago

@Steven Chase nice solution ,upvoted.

Talulah Riley - 6 months, 1 week ago

Travel to nearest star

The answer is 1464.

1 solution

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