How do you find the distance travelled from #0<=t<=1# by an object whose motion is #x=e^tcost, y=e^tsint#?

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Answer 1 · 2018-05-09T11:46:40

#sqrt 2 \ (e - 1)#

Let #z(t) = x(t) + i \ y(t) = e^((1+i)t)#

#dot z = (1+i) e^((1+i)t) #

Speed (not velocity) is needed to calculate distance #s#:

#s = int_0^1 sqrt (abs(dotz)^2) \ dt#

#abs(dotz)^2 = dot z bar (dot z)#

# = (1+i) e^((1+i)t) * (1-i) e^((1-i)t) = 2 e^(2t)#

#implies s =sqrt 2 int_0^1 e^(t) \ dt = sqrt 2 \ (e - 1)#