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Projectile motion of a particle

  1. Nov 21, 2004 #1
    Some particle is given an initial velocity, u at an angle θ to the horizontal. I'm asked to find (as a function of θ): 1. the range of the particle, X, 2. the maximum altitude reached, Y and the time taken to reach maximum altitude, T.

    First I resolved u into components: uy = usin θ, ux = ucos θ.

    For 1, I said x = ucos θ.t (since the horizontal motion is unaccelerated, no air resistance).

    To work out t, I used s = ut + (1/2)at^2 for the vertical motion, setting s = 0. For t != 0, I got t = (2usin θ)/g

    => X = (ucos θ.2usin θ)/g = (2u^2.sin θcos θ)/g = (u^2.sin 2θ)/g

    For 2, I used the fact that v = 0 when the particle reaches its maximum height and the equation v^2 = u^2 + 2as.

    => Y = (1/2g)(uy)^2 = (1/2g)(usin θ)^2

    For 3, I again used v = 0 at maximum height, but used v = u + at

    => T = (usin θ)/g

    Then, I'm asked to work out the projectile's velocity as a function of time.

    v(t) = [(vx)^2 + (vy)^2]^1/2

    vx = ucos θ
    Using v = u + at, vy = usin θ - gt

    v(t) = [(ucos θ)^2 + (usin θ - gt)^2]^1/2

    v(t) = [(ucos θ)^2 + ((usin θ)^2 - 2gtusin θ + (gt)^2]^1/2

    v(t) = [u^2.(cos^2 θ + sin^2 θ) - 2gtusin θ + (gt)^2]^1/2

    v(t) = [u^2 - 2gtusin θ + (gt)^2]^1/2

    Is this correct?
  2. jcsd
  3. Nov 21, 2004 #2

    Doc Al

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    Staff: Mentor

    Looks good to me.
  4. Nov 21, 2004 #3
    Cheers Doc! :)
  5. Nov 23, 2004 #4
    Hmm, I have to plot v(t) vs. t, for a given u and θ and all I get is a straight line. It doesn't seem right to me for some reason :(.
  6. Nov 23, 2004 #5

    Doc Al

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    Staff: Mentor

    Well, that can't be right. (I assume you are plotting the magnitude of the velocity.) Try it again! It starts out with its maximum value of u ... decreases to a minimum value of [itex]u cos\theta[/itex] (at the top of the motion)... then increases again back to the original value (when it's back to the starting height).
  7. Nov 23, 2004 #6

    Thanks again Doc. Yes, I'm plotting the magnitude of the velocity. My working for v is in my first post and I'm not sure what's wrong with it. Maybe I shouldn't have factorised u^2.cos^2 θ + u^2.sin^2 θ by u^2, but it shouldn't make a difference.
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