Calc 1 Honors Contract.docx .edu



Mason DenneyMAT265Prof. Brewer12 December 2012AbstractDuring 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.VariablesgGravity on the moon is 1.62 m/s^2 compared to earth’s 9.8 m/s^2thetathe angle of the jump is estimated at 45 degrees or (p/4)hiZero because jump occurs on a flat plane tt= (x/[vcos(theta)])t= x/(1.62cos(pi/4))vxyKinetic Energy =Potential Energy.5mvxy^2 = (mghof a jump)vxy= sqrt(2ghof a jump)vxy= sqrt(2*(1.62)*(.5))vxy=1.62 m/svysin(pi/4) = vy/vxyvy= vxy*sin(pi/4)vy= (1.62)*sin(pi/4) Height Equation from Integralsacceleration(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)xy= -.617284x^2+xy’=(2)(-.617284)x + 1 -1.234568x +1=0~max of y found at y’=0~-1.234568x = -1x= -1/(-1.234568)x= .81 meters~x is half of total range~Max height = -.617284x^2+xx=.81319786039370= -.617284*(.81)^2+(.81)= -.41 + .81= .4 mrange2x2(.81)1.62 metersVideo and Stills height grounded ................
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