Introduction
In this problem we were given two angles plus a velocity and tasked with finding the overall length that a skier traveled. There were a number of ways to complete this problem; I decided to integrate the position vector from a simple acceleration vector, then I found the total time which then gave me the total distance. The following are the given properties, distances are in meters and angles are in degrees.
Initial Velocity (v0) = 30
Acceleration in x-axis (ax) = 0
Acceleration in y-axis (ay) = -9.8
So the acceleration vector = (-- removed HTML --)
=12
=40 d= distance
Integrating to find velocity and position
First I derived velocity from the acceleration vector. I did this by breaking the vector into the x-axis and y axis. The velocity can be broken down using the first angle of 12 to give the starting velocity of (30 cos 12,30 sin 12)
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To correct for the landing slope, the starting position was broken down using the second angle. The tricky part is realizing how you set up your position. I made the starting position the inverse of the final position. Thus, x-axis position is negative and the y-axis is positive giving the starting position (-d cos 40,d sin 40)
-9.8t+30 sin (12) dt=-4.9t2+30t sin (12)+Cy1 y-axis integral
30 cos (12) dt=30t cos (12)+Cx1 x-axis integral
Cx1=-d cos 40 andCy1=d sin 40 Starting positions are the constants
p(t) = (-- removed HTML