Game Development Reference
In-Depth Information
Plugging this into our result from similar triangles, we have
v
s
p
r
=
,
v
s
= s∆t
lim
∆t→0
r ,
= s 2
v
∆t
lim
∆t→0
r .
(11.28)
The left-hand side of Equation (11.28) is a change in velocity over an in-
terval as the length of the interval approaches zero. This is the defini-
tion of instantaneous acceleration! Thus the magnitude of the acceleration
is s 2 /r.
Of course, acceleration is a vector quantity, and all we have determined
so far is its (constant) magnitude. What is the direction? To see this, com-
pare the vectors p (t) and ∆ v in Figure 11.15. Notice that they point in
opposite directions. In fact, in the limit as ∆θ goes to zero, they point in ex-
actly the opposite direction. That is, the acceleration is always towards the
center of the circle, which is why it is called centripetal (“center-seeking”)
acceleration.
Velocity and Acceleration of Uniform Circular Motion
When an object moves with constant speed s in a circular path with radius
r, the velocity v is tangent to the circle. The acceleration at any instant is
pointed towards the center of the circle and has magnitude
a = s 2 /r.
(11.29)
By combining some elementary geometry with some ideas of calculus,
we have obtained the most important facts about uniform circular motion.
A slightly different combination of geometry and calculus will yield the
actual kinematics equations. To this end, it will be helpful to refer to θ(t),
the angle that the vector p makes with with +x axis using the traditional
mathematical conventions, as shown in Figure 11.16.
Previously, we were concerned with the ∆θ, the change in this angle,
but now we consider its value as a function of time. We denote the initial
angle as θ(0) = θ 0 . We also define the angular frequency as ω = s/r, which
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