1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: A 3d Trajectory

  1. Mar 18, 2010 #1
    1. The problem statement, all variables and given/known data
    A particle of mass 1kg is projected in XYZ space, where Gravity (g=10m/s2) acts in -[tex]\hat{k}[/tex] direction. The initial velocity of a particle is [tex]\vec{u}[/tex]=(-3[tex]\hat{i}[/tex]+4[tex]\hat{j}[/tex])m/s.
    x-component of acceleration = 3t/4
    y-component of acceleration = -1 - 3t/4
    If total work done in interval t=0 to t=4 seconds is 90K Joules, then find the value of K.

    [The format of answering requires K to be an integer between 0 and 9 (inclusive)]


    2. Relevant equations
    Basic Kinematic definitions with complicated level of Calculus


    3. The attempt at a solution
     
  2. jcsd
  3. Mar 18, 2010 #2
    You should show some attempt at the solution. Without that we are not permitted to help.
    What about actually citing relevant equations and substituing them?
     
  4. Mar 18, 2010 #3
    What I tried was:

    ax = 3t/4
    vx = -3 + 3t2/4
    Fx = ma = 3t/4
    Px = Fx vx = -9t/4 + 9t3/32

    ay = -1 - 3t/4
    vy = -t + 3t2/8 +4
    Fy = ma = -1 - 3t/4
    Py = Fy vy = 9t3/32 + 9t2/8 - 2t - 4

    az = -10
    vz = -10t
    Fz = ma = -10
    Pz = Fz vz = 100t

    P = Px + Py + Pz = -9t/4 + 9t3/32 + 9t3/32 + 9t2/8 - 2t - 4 + 100t = 9t3/16 + 9t2/8 + 383t/4 - 4

    [tex]W = \int P dt[/tex]

    W = 9t4/64 + 3t3/8 + 383t2/8 - 4t

    Work Done form 0 sec to 4 sec = W(4)-W(0) = 9(4)4/64 + 3(4)3/8 + 383(4)2/8 - 4(4) = 36 + 24 + 766 - 16 = 810 = 90(9)

    Hence K=9[\b]


    Please tell if this is correct...
     
  5. Mar 19, 2010 #4
    you made a typo in vx, however your Px is okay.

    Your calculation is otherwise correct.
     
  6. Mar 19, 2010 #5
    Oh yes, it should be vx = -3 + 3t2/8

    But is there any smarter method which is less vulnerable to calculation errors?
     
  7. Mar 19, 2010 #6
    I used to avoid calculation errors by using a math package and always denoting the units.
    Your example in sympy:
    Code (Text):

    $ isympy
    Python 2.6.4 console for SymPy 0.7.0-git

    These commands were executed:
    >>> from __future__ import division
    >>> from sympy import *
    >>> x, y, z = symbols('xyz')
    >>> k, m, n = symbols('kmn', integer=True)
    >>> f, g, h = map(Function, 'fgh')

    Documentation can be found at http://sympy.org/

    In [1]: s,m,kg=symbols("s,m,kg",real=True,positive=True)

    In [2]: v0=Matrix([-3*m/s,4*m/s,0*m/s])

    In [3]: t=symbols("t",real=True,positive=True)

    In [4]: a=Matrix([3*t/4*m/s**3,-1.0*m/s**2-3*t/4*m/s**3,-10.0*m/s**2])

    In [5]: v=v0+integrate(a,(t,0,t))

    In [6]: F=a*1*kg
    In [7]: a
    Out[7]:
    ⎡   3⋅m⋅t    ⎤
    ⎢   ─────    ⎥
    ⎢       3    ⎥
    ⎢    4⋅s     ⎥
    ⎢            ⎥
    ⎢  3⋅m⋅t   m ⎥
    ⎢- ───── - ──⎥
    ⎢      3    2⎥
    ⎢   4⋅s    s ⎥
    ⎢            ⎥
    ⎢  -10.0⋅m   ⎥
    ⎢  ───────   ⎥
    ⎢      2     ⎥
    ⎣     s      ⎦

    In [8]: v
    Out[8]:
    ⎡               2  ⎤
    ⎢    3⋅m   3⋅m⋅t   ⎥
    ⎢  - ─── + ──────  ⎥
    ⎢     s        3   ⎥
    ⎢           8⋅s    ⎥
    ⎢                  ⎥
    ⎢                 2⎥
    ⎢4⋅m   m⋅t   3⋅m⋅t ⎥
    ⎢─── - ─── - ──────⎥
    ⎢ s      2       3 ⎥
    ⎢       s     8⋅s  ⎥
    ⎢                  ⎥
    ⎢    -10.0⋅m⋅t     ⎥
    ⎢    ─────────     ⎥
    ⎢         2        ⎥
    ⎣        s         ⎦

    In [9]: F
    Out[9]:
    ⎡    3⋅kg⋅m⋅t     ⎤
    ⎢    ────────     ⎥
    ⎢         3       ⎥
    ⎢      4⋅s        ⎥
    ⎢                 ⎥
    ⎢   ⎛  3⋅m⋅t   m ⎞⎥
    ⎢kg⋅⎜- ───── - ──⎟⎥
    ⎢   ⎜      3    2⎟⎥
    ⎢   ⎝   4⋅s    s ⎠⎥
    ⎢                 ⎥
    ⎢   -10.0⋅kg⋅m    ⎥
    ⎢   ──────────    ⎥
    ⎢        2        ⎥
    ⎣       s         ⎦

    In [10]: pp=[]

    In [11]: for i in range(3):  pp.append((F[i]*v[i]).expand())
       ....:

    In [12]: P=Matrix(pp)
    In [13]: P
    Out[13]:
    ⎡                        2         2  3               ⎤
    ⎢                9⋅kg⋅t⋅m    9⋅kg⋅m ⋅t                ⎥
    ⎢              - ───────── + ──────────               ⎥
    ⎢                      4           6                  ⎥
    ⎢                   4⋅s        32⋅s                   ⎥
    ⎢                                                     ⎥
    ⎢          2           2             2  2         2  3⎥
    ⎢  2⋅kg⋅t⋅m    4.0⋅kg⋅m    1.125⋅kg⋅m ⋅t    9⋅kg⋅m ⋅t ⎥
    ⎢- ───────── - ───────── + ────────────── + ──────────⎥
    ⎢       4           3             5               6   ⎥
    ⎢      s           s             s            32⋅s    ⎥
    ⎢                                                     ⎥
    ⎢                                2                    ⎥
    ⎢                    100.0⋅kg⋅t⋅m                     ⎥
    ⎢                    ─────────────                    ⎥
    ⎢                           4                         ⎥
    ⎣                          s                          ⎦

    In [14]: P_sum=P[0]+P[1]+P[2]

    In [15]: P_sum
    Out[15]:
                2           2             2  2         2  3
    95.75⋅kg⋅t⋅m    4.0⋅kg⋅m    1.125⋅kg⋅m ⋅t    9⋅kg⋅m ⋅t
    ───────────── - ───────── + ────────────── + ──────────
           4             3             5               6  
          s             s             s            16⋅s    

    In [16]: W=integrate(P_sum,t)

    In [17]: W
    Out[17]:
                2              2  2             2  3         2  4
      4.0⋅kg⋅t⋅m    47.875⋅kg⋅m ⋅t    0.375⋅kg⋅m ⋅t    9⋅kg⋅m ⋅t
    - ─────────── + ─────────────── + ────────────── + ──────────
            3               4                5               6  
           s               s                s            64⋅s    

    In [18]: Wsum=integrate(P_sum,(t,0*s,4*s))

    In [19]: Wsum
    Out[19]:
              2
    810.0⋅kg⋅m
    ───────────
          2    
         s    


     
     
  8. Mar 19, 2010 #7
    but i cant really use computer during exams!
     
  9. Mar 20, 2010 #8
    You can check units also by hand.
    And you can practice.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook