Using Cartesian Coordinates to Describe 2D Motion
Y
P1
y1
v1
R1
y2
P2
R2
j = (0,1)
motion path described
by the particle on 2D
v2
x1
i = (1,0)
x2
X
R
= x=ix(t)
+ y ij += y(t)
(x, 0)
(0, y)0)= +(x,(0,y)y(t)) = (x(t), y(t))
R(t)
j =+(x(t),
.
.
.
.
..
..
v(t) = R(t) = (x(t), y(t))  always in the tangent direction of the motion path
a(t) = v(t) = (x(t), y(t))
Using Polar Coordinates to Describe 2D Motion
Y
P1
v1
motion path described
by the particle on 2D
R1
eq
eq’ er
r1
q1
er’
R2
v2
X
radial acceleration
angle acceleration
R(t) = r(t) er(t)
 q information is already contained in er, so eq does not show up in R(t)
 If R1(t) = r1(t) er, R2(t) = r2(t) er’, R1(t) + R2(t)  (r1(t) + r2(t)) er
since er(t)  er’ (t). So, how do we do vector algebra in polar coordinates?
.
.
.
v(t) = R (t) = r(t) er(t) + r(t) er(t)
= vr(t) er(t) + vq(t) eq(t)  always in the tangent direction of the motion path
.
..
.
.
..
a(t) = v (t) = r(t) er(t) + 2r(t) er(t) + r(t) er(t) = ar(t) er(t) + aq(t) eq(t)
Using Tangential/Normal Coordinates to Describe 2D Motion
Using T/N coordinate, one is only interested in the velocity and acceleration of the
Y
moving particles. Hence, there is no position vector.
P1
motion path described
It is best to imagine you are traveling along the
by the1particle on 2D
motion path (see figure). T/N coordinate system
relies on the arc-length variable, S, to measure
the velocity and acceleration. Given any instant,
say t0, for any t > t0
P
S(t) measures the distance traveled by the particle
From t0 to t.
X
To define the speed, consider t > t0, the limit
•
It is no longer
S(t) = limtt0 S(t)/(t – t0)
required for a
defines
the velocity at point P1. Imagine the limit as if there are
fixed
origin.
t3 > t2 > t1 > … > t0 and measure the corresponding S(t3), S(t2),
S(t1), … In the limit tt0, S(t)  0 and t – t0  0. But its ratio
measures the speed of the velocity.
Using Tangential/Normal Coordinates to Describe 2D Motion
•
But, S(t) measures the speed, which is the magnitude
of the velocity. The direction of the velocity has not
yet been specified. So, where is it? Recall from Cartesian,
Polar coordinate systems, the velocity is always in the
tangent direction of the motion path.
P1
Therefore, the velocity expressed in T/N coordinate is
•
v(t) = S(t) et (t)
magnitude of
velocity
(tangential) direction
of the velocity
(see the red arrow)
en
et
The acceleration follows immediately by differentiation:
•
••
•
a(t) = v(t) = S(t) et(t) + S(t) et(t) = at(t) et(t) + an(t) en(t)
tangent acceleration measures
how fast S(t) changes in time.
normal acceleration, which describes the change in the direction of motion
and points toward the center of curvature at the given position of the
motion path (see the blue arrow).
Using Tangential/Normal Coordinates to Describe 2D Motion
Y
P1
en
et
motion path described
by the particle on 2D
en
P2
et
X