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Projectile - 2D motion - incline - acceleration

  1. Oct 6, 2015 #1
    1. The problem statement, all variables and given/known data
    A test rocket is fired from rest at point (A) along (b) a 200m incline, accelerating 1.25 m/s2
    2it0qv.png
    Given:
    b = 200 m, the angle at point A is 35°, a = 1.25 m/s2

    a) Find the maximum height above the ground.
    b) Find the greatest range.

    2. Relevant equations
    If it was a constant velocity I would have no problem solving this but I dont know how to deal with this acceleration. My book doesnt seem to help me, it is rated the most difficult in the chapter of 2-3D motions. but there are no examples with varing acceleration in the horizontal direction.

    When velocity is given
    v0y = v0 sinθ
    vy = v0y - gt1 = 0
    t1 = voy/g
    h = voyt1 - ½gt12

    vox = v0 cosθ
    y = 0 = voyt2 - ½gt22 = t2(voy - ½gt2)
    t2 = 2voy/g
    R = voxt2

    But how do I find these when its acceleration?

    3. The attempt at a solution
     
  2. jcsd
  3. Oct 6, 2015 #2

    Geofleur

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    Hint: Consider dividing the problem into two parts.
     
  4. Oct 6, 2015 #3
    does it make sense to find the final velocity at the 200 m mark on the ramp then calculate the height of the triangle and then go from there?
     
  5. Oct 6, 2015 #4

    BvU

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    Yes. That is splitting in two parts....
    And: you seem to think the horizontal acceleration is varying. Why ?
     
  6. Oct 6, 2015 #5
    Now that you say that it is very clear that it is! thank you so much <3
     
  7. Oct 7, 2015 #6

    BvU

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    So, just for the sake of other readers ( :wink: ), where are we now ?
     
  8. Oct 7, 2015 #7
    I went to bed, but today we're at this point:

    vf2 = vi2 + 2 * a * d
    vf = sqrt(0 + 2 * 1.25 m/s2 * 200 m) = 22.361 m/s

    The height of the triangle is, sin(35°)*200m = 114,7m = a

    -114.7m = (22.361 m/s * sin(35°))*t+(-½*9.8 m/s2 * t2)
    0 = -½*9.8m/s2*t2+sin(35°)*22.361m/s*t+114.7m
    Δt = (-sin(35°)*22.361m/s ±sqrt((sin(35°)*22.361m/s)-4*(-1/2*9.8)*114,7m)) / -9.8 m/s2
    Δt =+√ = -5,73s , Δt = -√ = 3.776s

    v0y = v0 sinθ
    vy = voy - gt1 = 0
    t1 = v0y/g
    h = v0yt1 - ½gt12

    v0y = sin(35°)*22,361 m/s = 12,83 m/s
    vy = 12,83 m/s - 9.8m/s2*t1 = 0
    t1 = voy/g = 12.83 m/s/9.8m/s2 = 1.3s
    h = v0yt1 - ½gt12 = 12,83m/s*1.3s-½*9.8m/s2*1.322 = 8,398m
    114,7m + 8.398m = 123,098m
    This doesnt make sense to me, 8 meters height from a rocket, I know there are different rockets I just expected more. (its a huge ramp for 8 meters of height)

    vox = 22.361 m/s * cos(35°) = 18,3 m/s
    y = 0 = voy * t2 - ½ * 9.8 m/s2 *t2 = t2(voy - ½ * g * t2)
    t2 = (2*voy)/9.8 m/s2 = 2,62s
    R = vox*t2 = 18,3 m/s * 2,62s = 47,96m
    but that length is still 114,7m above the ground.

    x = (v0*cos(-35°)*t = 22,361 m/s * cos(-35°)*3,776s = 67,17m
    47,96m + 67,17m = 117,125m

    Ok im pretty confused at this point and pretty sure I made a lot of mistakes, unfortunately.
     
  9. Oct 7, 2015 #8

    BvU

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    Oh boy, what a long post. And then I have to ask where the first line comes from (I don't see it in the relevant equations, that's why I ask :wink: not the I went to bed, but the vf2 line of course)
     
  10. Oct 7, 2015 #9
  11. Oct 7, 2015 #10

    BvU

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    Good. I've had some coffee since. Big four sounds impressive and is right here. Let me continue.... a bit slower :smile:

    Note that the d in the first one of these big four first is distance covered (*) so for your h that meant height counting from leaving the ramp....
    [edit] more coffee ⇒ yeah, 8 m counting from the top of the ramp. It's not a fast rocket, I admit...
    more like a radio-controlled toy car.
    All because 1.25 m/s2 isn't all that much.​


    (*) see ##
    x = x_0 + v_0 t + (1/2) a t^2\ ## here in our own big toolkit.
     
    Last edited: Oct 7, 2015
  12. Oct 7, 2015 #11
    that is so sad with the d :(
     
  13. Oct 7, 2015 #12
    also im a bit confused with this equation how would I find the initial velocity at the end/top of the ramp.
    all I know is x = 200 + 0 + 1/2 * 1.25 * t^2, I dont know either x or t
     
  14. Oct 7, 2015 #13

    BvU

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    At the end of the ramp you already found v = 22.36 m/s (vx = 18.32 m/s, vy = 12.82 m/s). From there you also found a flight time of 3.77 sec (I found 3.704, actually (using g = 9.8 m/s2) -- perhaps you can check; the other one was -6.32 s) ).

    From that point the horizontal motion is uniform, so distance covered is just ##v_{0,x} \, t##, your 67.2 m (my result was 67.8).

    I think you did well, but made it more complicated than necessary...
     
  15. Oct 7, 2015 #14
    I used v = 22,361 m/s vx = 18,3 m/s vy = 18,3 m/s flight time = 3,776s using g = 9,8 m/s2 the negative time had a larger difference compared to yours, I have -5,73s
     
  16. Oct 7, 2015 #15
    I think I made a mistake during the flight time calculation

    (-sin(35)*22.361-sqrt((sin(35)*22.361)-4(-1/2*9.8)*114.7))/-9.8 = 6.16s
    (-sin(35)*22.361+sqrt((sin(35)*22.361)-4(-1/2*9.8)*114.7))/-9.8 = -3,54s

    This is using a calculator I know how works, the other numbers was from Maple which i think always calculate in radians unless you do the 1/180*pi conversion
     
  17. Oct 7, 2015 #16

    BvU

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    Excel (I didn't do any rounding - except g = 9.8) says
    (PF won't use courier for code ? )

    Code (Text):


    10 h0 114.7153 =200*SIN(35/180*PI())
    11
    12
    13 v0^2 500 =2*1.25^1*200
    14
    15 v0 22.36068 =SQRT(D13)
    16
    17 v0x 18.3168 =D15*COS(35/180*PI())
    18 v0y 12.82556 =D15*SIN(35/180*PI())
    19
    20 t gnd 3.70366 =(0-D18+SQRT(D18^2+4*D10*9.8/2))/9.8
    21
    22 v0x * t gnd 67.83919 =D20*D17

     
     
  18. Oct 7, 2015 #17
    Yep I forgot to square b under the root.

    so my delta t's er now -3,7s and 6,32s

    Btw dont you have to divide by negative g and not positive?
     
  19. Oct 7, 2015 #18

    BvU

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    o:) Oops. Brilliant ! Sounds very reasonable to me :smile: and gets the projectile a little further as well. Perhaps we should top it off with a drawing showing the trajectory (y versus x) !

    [edit] Must have more coffee (again): at the launch from the top of the ramp back in time the parabola should be shorter than forward, so physical intuition should have kicked in and objected right away !
     
  20. Oct 7, 2015 #19
    As you can see from my earlier paint drawing, drawing is not a set of skills I was blessed with.

    And I guess the questions are done if I just fix the b^2 and correct the calculations using the time.
     
  21. Oct 7, 2015 #20
    vf2 = vi2 + 2 * a * d
    vf = sqrt(0 + 2 * 1.25 m/s2 * 200 m) = 22.361 m/s

    The height of the triangle is, sin(35°)*200m = 114,7m = a

    -114.7m = (22.361 m/s * sin(35°))*t+(-½*9.8 m/s2 * t2)
    0 = -½*9.8m/s2*t2+sin(35°)*22.361m/s*t+114.7m
    Δt = (-sin(35°)*22.361m/s ±sqrt((sin(35°)*22.361m/s)2-4*(-1/2*9.8)*114,7m)) / -9.8 m/s2
    Δt =+√ = -3,7s , Δt = -√ = 6,32s

    v0y = v0 sinθ
    vy = voy - gt1 = 0
    t1 = v0y/g
    h = v0yt1 - ½gt12

    voy = sin(35°)*22,361 m/s = 12,83 m/s
    vy = 12,83 m/s - 9.8m/s2*t1 = 0
    t1 = voy/g = 12.83 m/s/9.8m/s2 = 1.3s
    h = v0yt1 - ½gt12 = 12,83m/s*1.3s-½*9.8m/s2*1.322 = 8,398m
    114,7m + 8.398m = 123,098m
    This doesnt make sense to me, 8 meters height from a rocket, I know there are different rockets I just expected more. (its a huge ramp for 8 meters of height)

    vox = 22.361 m/s * cos(35°) = 18,3 m/s
    y = 0 = voy * t2 - ½ * 9.8 m/s2 *t2 = t2(voy - ½ * g * t2)
    t2 = (2*voy)/9.8 m/s2 = 2,62s
    R = vox*t2 = 18,3 m/s * 2,62s = 47,96m
    but that length is still 114,7m above the ground.

    x = (v0*cos(-35°)*t = 22,361 m/s * cos(-35°)*6,32s = 115,76m
    47,96m + 115,76m = 163,7m

    Something tells me this last part from x = ... is incorrect.
    Is 22,361 m/s * cos(-35°)*6,32s the range of the rocket from the top of the ramp and to the ground 114,7m below?
     
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