Two people, Paige and Shane, are racing each other. Assume that both their accel

Question

Two people, Paige and Shane, are racing each other. Assume that both their accelerations are constant, Paige covers the last 1/7 of the race in 5 seconds, and Shane covers the last 1/5 of the race in 7 seconds. Who wins, and by how much?

Using this formula, ---------we find that the time (in seconds) it takes Paige to run the race is----------- that the time it takes Shane is ---------- . Thus--------- wins the race by ------------seconds.

----------- mean *blank*, which is where I must put the answer.

Explanation / Answer

formula is
t= (2d/a)^(1/2)
let's look at the total time for each runner
let the race be d yards, and their acceleration be a


paige: covers the first 6/7 of the race in t1 seconds, then takes 5 seconds for the last 1/7

time for the first 6d/7 is: t1 = (12d/(7a))^(1/2) = (12/7)^(1/2)(d/a)^(1/2) (since this will be the part of the race beginning at rest)

sqrt(12/7) = 1.309
thus, total time for paige is: tD = 1.309(d/a)^(1/2) + 5


shane: covers the first (4/5) of the race in t2 seconds, then takes 7 seconds for the last 1/5
time for first 4d/5 is t2 = (8d/(5a))^(1/2) = (8/5)^1/2(d/a)^(1/2) = 1.265 (d/a)^(1/2)
total time for Paige is tP = 1.265 (d/a)^(1/2) + 7

there are two unknowns: total distance and acceleration

the difference in time will depend on the relation between these two
shanes time-Paige's time = (1.265-1.309)(d/a)^(1/2) + 2

let the times be equal:
(1.265 - 1.309)(d/a)^(1/2)+2 = 0
2= (1.309 - 1.265)(d/a)^(1/2)
(d/a)^(1/2) = 2 /(1.309 - 1.265) =45.45
d/a = 2066.12

unless the race is VERY long (or the acceleration VERY small), the 2 second difference in the last portion of the race will allow paige to win

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