# A 3d Trajectory

particlemania

## Homework Statement

A particle of mass 1kg is projected in XYZ space, where Gravity (g=10m/s2) acts in -$$\hat{k}$$ direction. The initial velocity of a particle is $$\vec{u}$$=(-3$$\hat{i}$$+4$$\hat{j}$$)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)]

## Homework Equations

Basic Kinematic definitions with complicated level of Calculus

## Answers and Replies

magwas
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?

particlemania
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?

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

$$W = \int P dt$$

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...

magwas
you made a typo in vx, however your Px is okay.

Your calculation is otherwise correct.

particlemania
Oh yes, it should be vx = -3 + 3t2/8

But is there any smarter method which is less vulnerable to calculation errors?

magwas
I used to avoid calculation errors by using a math package and always denoting the units.
Your example in sympy:
Code:
\$ 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 : s,m,kg=symbols("s,m,kg",real=True,positive=True)

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

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

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

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

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

In : pp=[]

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

In : P=Matrix(pp)
In : P
Out:
⎡                        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 : P_sum=P+P+P

In : P_sum
Out:
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 : W=integrate(P_sum,t)

In : W
Out:
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 : Wsum=integrate(P_sum,(t,0*s,4*s))

In : Wsum
Out:
2
810.0⋅kg⋅m
───────────
2
s

particlemania
but i cant really use computer during exams!

magwas
but i cant really use computer during exams!

You can check units also by hand.
And you can practice.