Vector Functions Amusement Park Ride

[rashomon]

Homework Statement

Kate's mother puts her on an amusement park ride. While on the ride, Kate follows the path
r(t) = (t-sin(t))i + (1-cos(t))j + 0 k for 0≤t≤2π. Kate's mother stands at location (2π, 4, 0)
while Kate is on the ride. Kate is a little scared, so she hangs on tight and stares straight ahead until the ride ends.

(a) When, if ever (and at what location), does Kate stare straight at her mother while on the ride?

(b) Calculate the arc length of Kate's ride as a function of time. How far does Kate travel on the ride?

(c) Using your arc length formula from part (b), how far would Kate go if she stayed on the ride for 0≤t≤4π? Comment on your results.

Homework Equations

r'(t)=<x'(t),y'(t),z'(t)>

The Attempt at a Solution

I am mostly troubled by part (a). I think it is the point at which the tangent to the space curve (r'(t)) would contain the given point if that tangent vector were extended into a line. But I can't seem to get very far with the numbers. Thanks.

 

[Mark44]

Homework Statement

Kate's mother puts her on an amusement park ride. While on the ride, Kate follows the path
r(t) = (t-sin(t))i + (1-cos(t))j + 0 k for 0≤t≤2π. Kate's mother stands at location (2π, 4, 0)
while Kate is on the ride. Kate is a little scared, so she hangs on tight and stares straight ahead until the ride ends.

(a) When, if ever (and at what location), does Kate stare straight at her mother while on the ride?

(b) Calculate the arc length of Kate's ride as a function of time. How far does Kate travel on the ride?

(c) Using your arc length formula from part (b), how far would Kate go if she stayed on the ride for 0≤t≤4π? Comment on your results.

Homework Equations

r'(t)=<x'(t),y'(t),z'(t)>

The Attempt at a Solution

I am mostly troubled by part (a). I think it is the point at which the tangent to the space curve (r'(t)) would contain the given point if that tangent vector were extended into a line. But I can't seem to get very far with the numbers. Thanks.

If the little girl is looking directly at her mother, her (daughter's) tangent vector will be parallel to the segment that joins the point on the ride with the point at which her mother is located.