Game Development Reference
In-Depth Information
(0 to t) v dt = ∫ (s1 to s2) ds
(0 to t) [v 1 / (1 + v 1 k t)] dt = ∫ (s1 to s2) ds
ln(1 + v 1 k t) / k = s 2 - s 1
If s 1 equals 0, then:
s = ln(1 + v 1 k t) / k
Note that in this equation, ln is the natural logarithm operator.
This example demonstrates the relative complexity of nonconstant acceleration prob‐
lems versus constant acceleration problems. It's a fairly simple example where you are
able to derive closed-form equations for velocity and displacement. In practice, however,
there may be several different types of forces acting on a given body in motion, which
could make the expression for induced acceleration quite complicated. This complexity
would render a closed-form solution like the preceding one impossible to obtain unless
you impose some simplifying restrictions on the problem, forcing you to rely on other
greater depth in Chapter 11 .
2D Particle Kinematics
When we are considering motion in one dimension—that is, when the motion is re‐
stricted to a straight line—it is easy enough to directly apply the formulas derived earlier
to determine instantaneous velocity, acceleration, and displacement. However, in two
dimensions, with motion possible in any direction on a given plane, you must consider
the kinematic properties of velocity, acceleration, and displacement as vectors.
Using rectangular coordinates in the standard Cartesian coordinate system, you must
account for the x and y components of displacement, velocity, and acceleration. Essen‐
tially, you can treat the x and y components separately and then superimpose these
components to define the corresponding vector quantities.
To help keep track of these x and y components, let i and j be unit vectors in the x- and
y-directions, respectively. Now you can write the kinematic property vectors in terms
of their components as follows:
v = v x i + v y j
a = a x i + a y j
If x is the displacement in the x-direction and y is the displacement in the y-direction,
then the displacement vector is:
s = x i + y j