A hawk flying at 19 m/s at an altitude of 165 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation `y = 165- x^2/57` until it hits the ground, where...
A hawk flying at 19 m/s at an altitude of 165 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation `y = 165- x^2/57`
until it hits the ground, where y is its height above the ground and x is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground.
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The distance equation is
`d = 165-1/57x^2`
This is a parabola with focal length `f=|-57/4| = 57/4`
We want to calculate the arclength from `x=0` to `x=sqrt((57)(165)) = 96.98`
Let `h = p/2` where `p` is the distance along the x-axis from the vertex to the point where we are measuring the arclength
`implies` `h = 96.98/2 = 48.49 `
` ` and let `q = sqrt(f^2+h^2) = 50.54`
Then the arclength `s` satisfies
`s = ((hq)/f) + fln((h+q)/f) = 199.6m` (look for length of an arc of a parabola in reference below)
Alternatively, evaluate `s = int_0^96.98 sqrt(1+((dy)/(dx))^2) dx` (this involves integrating `sec^3u` after making the substitution `4/57x = tanu`)
The prey travels 199.6m
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