You are at a point x=x(t) along a horizontal line, representing the ground. You are flying a kite which maintains a constant height of 40 meters. Assume also that the kite string is a straight line. The kite is above the point k=k(t) on the ground, which is currently 30 meters behind you.

Question

You are running to the right at 1m/sec; in other words, x′(t)=1 m/s. Also, the kite string is lengthening at 1 m/sec. What is the horizontal velocity of the kite k′(t) at this moment?

Answer 1

The length of the kit string is z(t). We have

z^2 = 40^2 + (x-k)^2
x' = 1
z' = 1
At the moment in question, x-k = 30, so z=50

2z z' = 2(x-k)(x'-k')
100 = 60(1 - k')
5/3 = 1 - k'
k' = -2/3

Seems odd, doesn't it? But a little consideration should convince you that it is so.