Game Development Reference
In-Depth Information
V
P
R
Figure 11.16
The position of the particle can be
identified by the angle θ
is measured in radians per second. 39 Thus, we can express the angle at any
given time as
Angle as a function of
time
θ(t) = θ 0 + ωt.
We've seen the parametric equation for a circle before in Section 9.1, so we
know how to express the kinematics equations for the particle's position in
terms of the radius r and the angle θ(t), as
x(t) = r cos(θ(t)) = r cos(θ 0 + ωt),
y(t) = r sin(θ(t)) = r sin(θ 0 + ωt).
Position as a function of
time
Since the velocity function is the derivative of the position function, we
can differentiate these equations to obtain the velocity equations. Luckily,
we learned the derivatives of the sine and cosine functions in Section 11.4.6
and the chain rule in Section 11.4.7. Differentiating gives us
x(t) = d
Velocity as a function of
time
dt (r cos(θ 0 + ωt)) = −rω sin(θ 0 + ωt),
y(t) = d
dt (r sin(θ 0 + ωt)) = rω cos(θ 0 + ωt).
39 Omega (ω) is the traditional letter for angular frequency. To see where the cal-
culation s/r comes from, consider that the circumference of the circle is 2πr, and this
distance is traversed at a speed of s. Therefore the angular frequency is 2πr/s revolu-
tions per second. But one revolution is equal to 2π radians, so the factor of 2π cancels
out. This is an example of why the use of radians is often so convenient (provided we
are working symbolically and aren't concerned with the numeric values of any angles).
 
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