Math Questions and Answers

Start Your Free Trial

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.

Expert Answers info

mathsworkmusic eNotes educator | Certified Educator

calendarEducator since 2012

write511 answers

starTop subjects are Math, Science, and Business

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

check Approved by eNotes Editorial