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Kinematics: Relative-Motion Analysis of Two Particles Using Translating…
Kinematics: Relative-Motion Analysis of Two Particles Using Translating Axes
Particle Kinematics:
Relative Position
The
absolute position
of two particles A and B wrt the fixed x,y,z reference frame are given by vectors
r(subscript A)
and
r(subscript B)
. The
position of B relative to A
is represented by:
r(subscript B/A)
=
r(subscript B)
-
r(subscript A)
We've
translated
our fixed axis (x,y,z) onto a chosen (moving) reference point (x',y',z'). Here, A becomes the fixed reference point from which we can measure B
Therefore, if
r(subscript B)
= (10
i
+ 2
j
)m and
r(subscript A)
= (4
i
+ 5
j
)m, then
r(subscript B/A)
= (6
i
- 3
j
)m
Particle Kinematics:
Relative Velocity
To determine the
relative velocity
of B wrt A, the time derivative of the relative position equation is taken:
v(subscript B/A
=
v(subscript B)
-
v(subscript A)
OR
v(subscript B
=
v(subscript A)
+
v(subscript B/A)
In The below equations,
v(subscript B)
and
v(subscript A)
are called the
absolute velocities
and
v(subscript B/A)
is the
relative velocity
of B wrt A
Relative
Acceleration
The time derivative of the relative velocity equation yields a similar
vector
relationship between the
absolute
and
relative accelerations
of particles A and B
These derivatives yield:
a(subscript B/A)
=
a(subscript B)
-
a(subscript A)
OR
a(subscript B)
=
a(subscript A)
+
a(subscript B/A)
Relative Motion:
Solving Problems
Since the relative motion equations are
vector equations
, problems involving them may be solved in one of two ways
The velocity vectors in
v(subscript B)
=
v(subscript A)
+
v(subscript B/A)
could be written as 2D
Cartesian Vectors
and the resulting 2D scalar component equations solved for up to two unknowns
Vector problems could be solved
"graphically
by use of trigonometry. This approach makes use of
law of sines
or the
law of cosines
Revision:
Law of Sines and Cosines
Law of Sines
:
a / sin(A) = b / sin(B) = c / sin(C)
Law of Cosines
:
a^2 = b^2 + c^2 - 2bcCosA
b^2 = a^2 + c^2 - 2acCosB
c^2 = a^2 + b^2 - 2bcCosC
