Word problem

Question

If you and a friend are running a circular track and it takes you 12 minutes to complete a lap and it takes him 8 minutes to complete a lap, at what time will he pass you? If we run for 90 minutes at what other times will pass me again?

Answer 1

T₁ = time it takes slow runner to complete lap = 12 min

T₂ = time it takes fast runner to complete lap = 8 min

 

ω₁ = angular speed of slow runner = 2π/T₁

ω₂ = angular speed of fast runner = 2π/T₂

 

They meet at times t whenever the angles ωt swept out by their running differ by exactly 2πn radians, where

n = 0, 1, 2, ...

 

(2π/T₂)t - (2π/T₁)t = 2πn

 

t(1/T₂ - 1/T₁) = n

 

t = [T₁T₂/(T₁ - T₂)] n = [(12)(8)/(12 - 8)]n min = 24n min

n = 0, 1, 2, ...

 

n = 0:

 

They start together

 

n = 1:

 

t = 24 min, first passing

 

n = 2:

 

t = 2(24) min = 48 min, second passing

 

n = 3:

 

t = 3(24) min = 72 min, third passing

 

t = 4(24) min = 96 min > 90 min