Calculating Cross-Country Ski Distance

[ios]

A cross-country skier skis s1 in the direction theta1 west of south, then s2 in the direction theta2 north of east, and finally s3 in the direction theta3 south of west.

How far is the skier from the starting point? I drew it..and it looks really really weird, I haven't done physics in a couple years now, and I'm a bit rusty, any help on how to start this?

 

[HallsofIvy]

I don't know how it could "look wierd"- that would depend on what the numbers involved are! In general, it is a zig-zag path.

There are several different ways to do this. The way I would do it is to reduce to "components". Taking the positive x-component east and positive y-component n,
The components of a vector "of length s in direction θ (measured from due e) would be < r cos(θ), r sin(θ)>. The only problem is the angles.

If θ₁ is "west of south" the angle measured from east is 270- θ₁ and the vector is <s₁cos(270-θ₁), s₁sin(270-θ₁>.
If θ₂ is measured "north of east", then it is already correct. The vector is <s₂cos(θ₂),s₂sin(θ₂)>.
If θ₃ is measured "south of west", then we need to add 180 degrees to get the angle measured from east. The vector is <s₃cos(θ₃+ 180),s₃sin(θ₃+ 180)>

Adding those components, the vector from the intial position to the final position is
<s₁cos(270-θ₁)+s₂cos(θ₂)+s₃cos(θ₃+ 180), s₁sin(270-θ₁+s₂sin(θ₂)+s₃sin(θ₃+ 180)>

To find the distance from the initial position to the final position, use the Pythagorean formula: square the two components, add and then take the square root.

If you really are given only "s₁, &theta;₁", etc. then you will have a very complicated expression. If you are given specific numbers, substitute those first and then calculate.

 

[ios]

i'm only given s1, theta1, s2, theta2, s3, theta3