Projectile Motion using Vector Calculus

Question

Trebuchet can throw an 80 lb rock up to 100 yards. The maximum height that the rock reaches as it flies through the air toward a fortress wall 100 yards away is 20 yards, calculate the initial speed and the angle.

Answer 1

y = a(x - h)² + k

y = a(x - 50)² + 20

0 = a50² + 20

-20 = 2500a

-1/125 = a

y = -1/125(x - 50)² + 20

dy/dx = -2/125(x - 50)

at x = 0, dy/dx = 0.8

tan θ = 0.8

θ = 38.6598º

Answer 2

height function H(x)

H(0)=0

H(100) = 0

max occurs at the average of the solutions, which is x=50.

So H(50) = 20 and H'(50)=0

H(x) = Kx(x-100)

20 = k(50)(-50)

k = 20/-2500 = -2/250 = -1/125

H(x) = (-1/125)x (x-100)

Note that H'(x) = (-1/125)x + (-1/125)(x-100)

= (-1/125) (x + x - 100)

= (-1/125)(2x - 100)

= (-2/125)(x-50)

at which the maximum is x=50

the angle for x=50 and y= 20 is

the inverse sine of 20/50 = 2/5

inverse_sine(2/5) = 23 and 4/7 degrees

The initial velocity is zero