Calc 1 Honors Contract

Mason Denney MAT265 Prof. Brewer 12 December 2012

Abstract During the Apollo missions to the moon, astronauts discovered an easier way to travel: bunny hopping. In the absence of air resistance, the bunny hops should compare to parabolic arches. I integrated the acceleration of gravity twice to find a height formula. I then plugged in the variables I made. I took the derivative of this equation and set it to zero to find x value at the maximum. By plugging in the found x value into the original function, I found the max height. I also found the range by multiplying the x value at the maximum by 2.

Variables g Gravity on the moon is 1.62 m/s^2 compared to earth’s 9.8 m/s^2 theta the angle of the jump is estimated at 45 degrees or (p/4) hi Zero because jump occurs on a flat plane t t= (x/[vcos(theta)]) t= x/(1.62cos(pi/4)) vxy Kinetic Energy =Potential Energy

.5mvxy^2 = (mghof a jump)

vxy= sqrt(2ghof a jump)

vxy= sqrt(2*(1.62)*(.5))

vxy=1.62 m/s vy sin(pi/4) = vy/vxy

vy= vxy*sin(pi/4)

vy= (1.62)*sin(pi/4)

Height Equation from Integrals acceleration(y’’) = -g ~Take integral use v instead of C~

velocity(y’) = -gt + vy w ~Take integral use hi instead of C & Replace vy~

Height (y) = (1/2)-(g)(t^2) + vxy(t) sin(theta) + hi

Plugging in variables to find y then deriving y’

Height (y) = (1/2)-(g)(t^2) + vxy(t) sin(theta) + hi

= (1/2)-(1.62)(t^2) + (1.62)(t) sin(pi/4) + 0 = (-1.62/2)*(t^2) + (1.62)*(t) sin(pi/4) = (-.81)*[(x/[1.62cos(pi/4)])^2] + (1.62)*(x/(1.62cos(pi/4))) sin(pi/4) = -.617284x^2+(sin45/cos45)x y = -.617284x^2+x y’ =(2)(-.617284)x + 1 -1.234568x +1 =0 ~max of y found at y’=0~ -1.234568x = -1 x= -1/(-1.234568) x= .81 meters ~x is half of total range~

Max height = -.617284x^2+x x=.81 = -.617284*(.81)^2+(.81) = -.41 + .81 = .4 m range 2x 2(.81) 1.62 meters

Video and Stills http://www.youtube.com/watch?v=HKdwcLytloU

max height grounded