# A hawk flying at 19 m/s at an altitude of 165 m accidentally drops its prey. How far does the prey travel if its trajectory is a parabola?The parabolic trajectory of the falling prey is described by the equation `y = 165 - 165/57x^2` 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.

The distance travelled is the arclength of the parabola `y = 165-165(x^2/57)` between `x=0` and the place where the prey hits the ground.

The prey hits the ground when `y=0`

ie when `x =sqrt(57)`

Write the parabola in terms of the parameter t:

`x = 2at + k`

`y = at^2 + h`

Since ` ``(y-h) = (x-k)^2/(4a)`

`implies a = -57/(4(165)) = -0.086`  `k= 0` and `h =165`

The arclength of a parabola is given by

`s = a(tsqrt(1+t^2) +sinh^(-1)t)`

When ` `` ``x = sqrt(57)` , `t = sqrt(57)/(2a) = -sqrt(57)/(2*0.086) = -43.71`

`implies s = 165.43 m`

The prey travels 165.43 metres parabolically