Captain Kirkís starship leaves Earth for Planet X at the same time Marvin the Martianís starship leaves Planet X for Earth. Each ship travels at a constant velocity, but one is faster than the other. After meeting and passing, Kirk requires 22.5 hours to reach Planet X, while Marvin requires only 10 hours to reach Earth. Exactly what total time did each starship require for its interplanetary journey? Assume stationary planets.
Credit for this problem goes to Cliff Pickover
call h the amount of time before they crossed paths.
Marvin's total time is 10+h and Kirk's is 22.5+h
both travelled distance d, so
Marvin's rate is d/(10+h) and Kirk's is d/(22.5+h)
In the first h hours, they covered the entire distance between them, so we have
d*h/(10+h) + d*h/(22.5+h) = d (d is not zero so the d's cancel)
h/(10+h) + h/(22.5+h) - 1 = 0 (common denomiators)
[22.5h + h^2 + 10h + h^2 - (225 + 32.5h + h^2)]/[225+32.5h+h^2] = 0 (this denomiator is irrelevant unless h=0 which only happens at -10 and -22.5 So simplify to get
h^2 - 225 = 0
h=15 (discard h=-15 because Kirk would have negative time)
Total times: Marvin= 25 hours, Kirk = 37.5 hours.
Curiously, we know neither the distance nor the rates.
Posted by Jer
on 2005-05-05 18:00:24