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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.

Question ID
524319

Created
April 2, 2011 8:12pm UTC

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0

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https://questions.llc/questions/524319

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2

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404

2 answers

  1. 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.

    Answer ID
    524333

    Created
    April 2, 2011 9:03pm UTC

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

    Answer ID
    524546

    Created
    April 3, 2011 4:10am UTC

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