Posted by **andrew** on Saturday, April 2, 2011 at 4:12pm.

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 -
**Damon**, Saturday, April 2, 2011 at 5:03pm
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 -
**andrew**, Sunday, April 3, 2011 at 12:10am
@Damon : Appreciate it sir. u are a life saver. i was on track and needed a kick . thanks alot.

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