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

solution techniques like numerical integration. We'll talk about this sort of problem in

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