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Homework Help: Projectile Motion in Bigger Radius.

  1. Nov 13, 2008 #1
    1. The problem statement, all variables and given/known data
    A projectile of mass, m is fired from the surface of the earth at an angle alpha from the vertical. THe initial speed v0 is equal to (GM/R)^0.5 where G is gravitational cont, M is mass of earth and R is radius of the earth. How high does the projectile rise? Neglect air resistance and the earth's rotation. r.max=r max in picture.

    http://img370.imageshack.us/my.php?image=quesgf6.jpg

    http://img370.imageshack.us/img370/189/quesgf6.jpg [Broken]
    http://g.imageshack.us/img370/quesgf6.jpg/1/ [Broken]


    2. Relevant equations

    U + K = constant


    3. The attempt at a solution

    So I started with that equation, and find out that
    -(GMm/R) + .5(m)(v0^2) = -(GMm/(r.max)) + .5(m)(vf^2)

    and from here we can eliminate the vf because it is zero when it is a the highest position. But after substitution that v0 = (GM/R)^0.5 this equation will become

    -(GMm/R) + .5(GMm/R) = -(GMm/(r.max))

    this will become r.max =2R which i think is impossible.

    Can anyone out there here me ? I am totally clueless. From the lecturer I found out that is it max when alpha is = 60degrees and the r.max is (3R/2) .

    Thanks in advance
     
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Nov 13, 2008 #2
    loss in Ke = Gain in GPE

    h = distance from the centre of the earth

    (1/2)m(v0 cos a)^2 = GMm/R - GMm/h

    (1/2)(m)(GM/R) (cos a)^2 = GMm ( 1/R - 1/h)

    Cancelling m and GM on both sides,

    (1/2R) (cos a)^2 = 1/R - 1/h

    1/h = 1/R - (1/2R) (cos a)^2

    1/h = (2 - cos^2 a)/2R

    h = (2R)/(sin^2 a + 1)


    Therefore, for a maximum h, sin a must be as small as possible, setting alpha = 0 degrees gives h = 2R.

    when alpha = 60 degrees, h = (2R)/(1.75) = 1.14R
     
    Last edited: Nov 13, 2008
  4. Nov 13, 2008 #3
    hey unscientific as I know that from the lecturer it is a round number.. Could it be I have done wrong somewhere? I got the same answer as well but I just want to confirm again?

    Anyone else might wanna try to help me?

    Thanks..
     
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