calculus

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1. Consider a projectile, say a frictionless teddy bear, which is thrown at an angle of theta with an initial
velocity of v0.

 a) For a fi xed angle, and the maximum height of the teddy bear.
 b) Calculate the distance that the teddy bear travels before returning (gently, of course) to the
ground (which is assumed to be
at).
 c) Use these pieces of information to nd the angle which produces the maximum of the sum of height and distance.
usage of function. in these question.

• calculus - ,

Vi = Vo sin T
u = Vo cos T

v = Vi - gt
max height when v = 0
0 = Vosin T - g t
t = Vo sin T / g
h = 0 + Vi t - .5 g t^2
max h = Vo sin T [Vo sin T/g] - .5 g[Vo sin T / g]^2
max h = [.5 /g][VosinT]^2
(this is quicker to get using potential and kinetic energy argumenents. Note max height of course when T = 90 degrees, straight up)

Now part b
the total time in the air is twice the time needed to reach max altitude so
total t = 2 Vo sin T/g
d = u t = Vo t cos T
d = Vo [ 2 Vo sinT/g]cos T
d = [2 Vo^2/g] sin T cos T

part c
Sum = S = [.5 /g][VosinT]^2 + [2 Vo^2/g] sin T cos T

dS/dT = 0 at max
0= [.5Vo^2/g]sin Tcos T+[2Vo^2/g][-sin^2T+cos^2T]
0=2 cos^2T -.5 sin T cosT -2 sin^2T
but 2 (cos^2T-sin^2T)=2 cos2T
0= 2 cos 2T -.5 sinTcosT
but sinTcosT =.5 sin2T
so
0=2 cos2T -.25 sin2T
sin2T/cos2T = 8 = tan 2T
2T = 82.8 degrees
T = 41.4 degrees
Interesting, max range is at 45 degrees. I may have made an arithmetic error in that mess.

• calculus - ,

@Damon : Appreciate it sir. u are a life saver. i was on track and needed a kick . thanks alot.