︠fcd10ac7-3b6b-46b7-a338-ab636866cd46s︠ var('x accel t c') # "x" is the midpoint distance to be travelled. c = 299792458 #light speed, in m/s ly = 9.4605284E15 #light year, in meters AU = 1.4960E11 #astronomical unit, meters assume(t>0) #time in seconds, divided by "units" for display. accel = 9.81*1.0 #acceleration, in m/s^2 #units=3600 #hours #units=3600*24 #days #units=3600*24*7 #weeks units=3600*24*365.25 #years #Midpoint is halfway to Mars (or wherever) http://nssdc.gsfc.nasa.gov/planetary/factsheet/marsfact.html #midpoint_distance = 4.01E11 / 2 #Mars farthest distance is 401 million km = 4.01E11 meters. #midpoint_distance = 1.00E11 / 2 #Mars close approach at 100 million km = 1E11 meters. #midpoint_distance = 0.55E11 / 2 #Mars close approach at 55 million km = 0.55E11 meters. #midpoint_distance = 30.3*AU / 2 #Neptune aphelion: https://en.wikipedia.org/wiki/Neptune #midpoint_distance = 83*AU / 2 #2012 VP113 distance from Sun in 2015: https://en.wikipedia.org/wiki/2012_VP113 #midpoint_distance = 86*AU / 2 #Sedna distance from Sun in 2015: https://en.wikipedia.org/wiki/90377_Sedna #midpoint_distance = 96.4*AU / 2 #Eris distance from Sun in 2014: https://en.wikipedia.org/wiki/Eris_(dwarf_planet) #midpoint_distance = 2000*AU / 2 #Inner edge of inner Oort cloud: https://en.wikipedia.org/wiki/Oort_cloud#Structure_and_composition #midpoint_distance = 20000*AU / 2 #Inner edge of outer Oort cloud: https://en.wikipedia.org/wiki/Oort_cloud#Structure_and_composition #midpoint_distance = 50000*AU / 2 #Outer edge of outer Oort cloud: https://en.wikipedia.org/wiki/Oort_cloud#Structure_and_composition #midpoint_distance = 0.0636*ly / 2 #Max speed at 1g is 0.25c, takes 0.49 ship years total. #midpoint_distance = 0.3*ly / 2 #Max speed at 1g is 0.5c, takes 1.06 ship years total. #midpoint_distance = 1.128*ly / 2 #Max speed at 1g is 0.775c, takes 2.00 ship years total. #midpoint_distance = 2.51*ly / 2 #Max speed at 1g is 0.9c, takes 2.85 ship years total. #midpoint_distance = 11.799*ly / 2 #Max speed at 1g is 0.99c, takes 5.13 ship years total. #midpoint_distance = 41.383*ly / 2 #Max speed at 1g is 0.999c, takes 7.36 ship years total. #midpoint_distance = 135.02*ly / 2 #Max speed at 1g is 0.9999c, takes 9.59 ship years total. midpoint_distance = 4.25*ly / 2 #Proxima Centauri. #midpoint_distance = 4.32*ly / 2 #Alpha Centauri. #midpoint_distance = 100.0*ly / 2 #midpoint_distance = 30000*ly / 2 #Sag A, center of Milky Way. #midpoint_distance = 2.5E6*ly / 2 #Andromeda galaxy. https://en.wikipedia.org/wiki/Andromeda_Galaxy #midpoint_distance = 93E9*ly / 2 #Diameter of the observable universe: https://en.wikipedia.org/wiki/Observable_universe #midpoint_distance = 1E12*ly / 2 #Ludicrously far. ~53 total ship years of travel at 1g. #midpoint_distance = 1E23*ly / 2 #Gone to plaid. ~103 total ship years of travel at 1g. See https://twitter.com/DumbSci/status/656296561882697728 (Ascension was still interesting, aside from that. Sorry for using big "G" rather than the correct little "g" in that tweet.) #Newtonian mechanics: eq1 = x == 0.5*accel*t^2 soln1 = solve(eq1.subs(x=midpoint_distance),t) accel_time = 2*soln1[0].rhs().n() accel_time/units #At high speeds Newtonian mechanics will yield absurd results. ︡fa943e7b-06e9-490e-9767-d7803a9ecf27︡︡{"stdout":"(x, accel, t, c)\n","done":false}︡{"stdout":"4.05736134029145\n","done":false}︡{"done":true} ︠c3b70281-1da0-48c0-a12b-3c7ad0b19e84s︠ #Maximum speed at midpoint, in m/s and fraction of "c". accel*accel_time/2 accel*accel_time/2/c #At high speeds Newtonian mechanics will yield absurd (>1) results. ︡049966e0-91da-48e9-b664-599e62784f54︡︡{"stdout":"6.28039075469831e8\n","done":false}︡{"stdout":"2.09491285958178\n","done":false}︡{"done":true} ︠711df174-5be2-4494-b112-ac0f4e3cd7a4s︠ #Relativistic travel time, as measured on the ship. # http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html # http://www.webcitation.org/query?url=http://www.geocities.com/albmont/relroket.htm&date=2009-10-25+12:17:16 var('accel_time_rel') eq2 = accel_time_rel == (c/accel)*arccosh(accel*midpoint_distance/c^2 + 1) soln2 = solve(eq2,accel_time_rel) accel_time_ship = 2*soln2[0].rhs().n() accel_time_ship/units ︡507c9a14-4410-42c5-bab3-14f0a388aeca︡︡{"stdout":"accel_time_rel\n","done":false}︡{"stdout":"3.54248691970821\n","done":false}︡{"done":true} ︠134a2275-ba39-49bf-9813-7b39217c5e3as︠ #Relativistic travel time, as measured on Earth. accel_time_earth = 2*(c/accel)*sinh(accel*accel_time_ship/2/c) accel_time_earth/units ︡8f11f4b2-e44a-491f-b46d-31060a4e0326︡︡{"stdout":"5.87570501785570\n","done":false}︡{"done":true} ︠c21cc830-9ac2-45d4-92dd-87f93fdfc33fs︠ #Maximum speed at midpoint, in m/s and fraction of "c". c*tanh(accel*accel_time_ship/2/c) beta = tanh(accel*accel_time_ship/2/c) beta ︡38b7fbcd-2131-47b7-b162-63b08b854cd2︡︡{"stdout":"2.84723419309041e8\n","done":false}︡{"stdout":"0.949735097435443\n","done":false}︡{"done":true} ︠dc6c826c-bfbc-4db5-be0b-1e64601b06c2s︠ #If engines are shut off to cruise at maximum speed, how long will it take to travel cruise_distance, as measured on Earth? #Note: total_distance can't be smaller than twice the midpoint_distance. #total_distance = 4.25*ly total_distance = 20*ly #total_distance = 30000*ly cruise_distance = total_distance - 2*midpoint_distance cruise_distance/ly #First display cruise distance in light years. cruise_time_earth = cruise_distance/(beta*c) cruise_time_earth/units ︡1be4ee9b-9013-4d02-805c-6e19796d8b26︡︡{"stdout":"15.7500000000000\n","done":false}︡{"stdout":"16.5832174012337\n","done":false}︡{"done":true} ︠41c3289e-9508-4e2b-aab7-3f442a64dd58s︠ #How long will it take to travel cruise_distance, as measured on ship? cruise_time_ship = cruise_time_earth*sqrt(1-beta^2) cruise_time_ship/units ︡b40dda8c-5991-4d8f-a937-9aa0d7109818︡︡{"stdout":"5.19145413552563\n","done":false}︡{"done":true} ︠eda75761-1ca2-4713-b189-f6618a36eaecs︠ #Total travel time as measured on the ship: (accel_time_ship+cruise_time_ship)/units ︡a764c4c2-8712-4bef-84b8-5f21854ffedb︡︡{"stdout":"8.73394105523384\n","done":false}︡{"done":true} ︠ab4da5ec-3d85-4358-8a3c-ee8cd5e97f7es︠ #Total travel time as measured on Earth: (accel_time_earth+cruise_time_earth)/units ︡33b5d52c-5260-4e06-8e7b-08bcf3596d31︡︡{"stdout":"22.4589224190894\n","done":false}︡{"done":true} ︠9cf62342-dc19-4e0f-b18e-ddc402079b60s︠ #Plot time (in "units") measured on the ship while accelerating and decelerating versus distance travelled in light years. #(Doesn't include time spent cruising at constant velocity.) T(x) = 2*(c/accel)*arccosh(accel*x*ly/2/c^2 + 1)/units plot(T,(x,0.1,10),axes_labels=['light years','ship years']) ︡7e06ae21-2e19-4b53-b25a-ac61ad1b0ed4︡︡{"once":false,"done":false,"file":{"show":true,"uuid":"4ae44642-8e6d-4d48-996a-1b1b676d81e5","filename":"/projects/9177c234-72f8-4cd0-b44a-11367a3bad6d/.sage/temp/compute2-us/2495/tmp_8AUSHR.svg"}}︡{"html":"
","done":false}︡{"done":true} ︠5b5577c9-c6ff-4e56-8d45-710650addbd1︠