Game Development Reference

In-Depth Information

Newton's Second Law of Motion

As we stated in the first section of this chapter, Newton's second law of motion is of

particular interest in the study of mechanics. Recall that the equation form of Newton's

second law is:

F = ma

where
F
is the resultant force acting on the body,
m
is the mass of the body, and
a
is the

linear acceleration of the body center of gravity.

If you rearrange this equation as follows:

F/m = a

you can see how the mass of a body acts as a measure of resistance to motion. Observe

here that as mass increases in the denominator for a constant applied force, then the

resulting acceleration of the body will decrease. You could say that the body of greater

mass offers greater resistance to motion. Similarly, as the mass decreases for a constant

applied force, then the resulting acceleration of the body will increase, and you could

say that the body of smaller mass offers lower resistance to motion.

Newton's second law also states that the resulting acceleration is in the same direction

as the resultant force on the body; thus, force and acceleration must be treated as vector

quantities. In general, there may be more than one force acting on the body at a given

time, which means that the resultant force is the vector sum of all forces acting on the

body. So, you can now write:

∑
F
= m
a

where
a
represents the acceleration vector.

In 3D, the force and acceleration vectors will have
x
,
y
, and
z
components in the Cartesian

reference system. In this case, the component equations of motion are written as follows:

∑ F
x
= ma
x

∑ F
y
= ma
y

∑ F
z
= ma
z

An alternative way to interpret Newton's second law is that the sum of all forces acting

on a body is equal to the rate of change of the body's momentum over time, which is

the derivative of momentum with respect to time. Momentum equals mass times ve‐

locity, and since velocity is a vector quantity, so is momentum. Thus: