Chapter 12:
Mechanics 2: Linear & Rotational Dynamics
Fletcher Dunn
Ian Parberry
Valve Software
University of North Texas
3D Math Primer for Graphics & Game Development
What You’ll See in This Chapter
This chapter considers the cause of motion, its
orientation, and how we might go about simulating it on
a computer. It is divided into six sections.
• Section 12.1 gives an overview of Newton’s 3 laws.
• Section 12.2 talks about the cause of motion: the force.
• Section 12.3 introduces momentum.
• Section 12.4 looks at collisions and impulse.
• Section 12.5 is about rotational dynamics.
• Section 12.6 discusses digital simulation of mechanics.
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Word Cloud
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Section 12.1:
Newton’s 3 Laws
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Sir Isaac Newton
• Sir Isaac Newton established
three simple laws that
provide a framework, which
we call Newtonian or
classical mechanics.
• It doesn’t hold at high speeds
or small distances, but it’s
good enough for everyday
life, and video games.
• (Image from Wikimedia
Commons.)
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Newton’s First Law
Every body persists in its state of being at rest
or of moving uniformly straight forward, except
insofar as it is compelled to change its state by
force impressed.
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Newton’s Second Law
The acceleration 𝐚 of a body is proportional to
(and in the same direction as) the net external
force 𝐟 acting on the body, and inversely
proportional to the mass 𝑚 of the body:
𝐟 = 𝑚𝐚.
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The Force
• Force is a vector. It has units like kg.m/sec2,
also called a Newton.
m
m
𝐟 = 𝑚 kg . 𝐚
= 𝑚𝐚 kg
2
sec
sec 2
• “Duct tape is like the force. It has a light side, a
dark side, and it holds the universe together.”
(Carl Zwanzig)
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Free Body Diagram
Starting with a representation of the object.
1. Draw and label all the forces acting on it.
2. Sum those forces (using vector addition) to
compute the net force.
3. Use Newton's 2nd law to compute the
acceleration of the object.
4. Integrate the acceleration to determine the
motion of the object.
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Differential Equations
When solving problems analytically, this means
solving differential equations. We don't use any
differential equations in this book because there
are only a few simple cases that we will look at
analytically. Numerical methods of integration
must be used. Later, we examine Euler
integration, which is the most simple method
imaginable, but also the one used by most realtime rigid body simulators.
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Inertial Reference Frames
• This only works in a reference
frame that is not accelerating.
• You have to invent fictional forces
to explain why objects are not
accelerating according to
Newton’s 1st and 2nd laws.
• A robot in a falling elevator is in a
noninertial frame. He must invent
a fictitious upward force to
counteract gravity to explain why
his herring sandwich doesn’t fall.
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To a passing alien who is not accelerating,
Newton’s laws work just fine, and there is no
need to invent a fictional force.
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Newton’s Third Law
To every action there is always an equal and
opposite reaction. Or, the forces of two bodies
on each other are always equal and are
directed in opposite directions.
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Example
There are four forces here.
1. Moe pushing the box.
2. The box pushing Moe.
3. Moe pushing the Earth.
4. The Earth pushing Moe.
Note that 𝐟𝐸𝑎𝑟𝑡ℎ and 𝐟𝐵𝑜𝑥
cancel out.
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Consequence of Newton’s 3rd Law
• As long as the internal forces cancel out, we
are justified in representing a complex body
by a single point or particle.
• This is called rigid body dynamics.
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Section 12.2:
Some Simple Force Laws
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Gravity in the Real World
Newton’s Law of Universal Gravitation:
𝑚1 𝑚2
𝑓=𝐺
𝑑2
where 𝑓 is the magnitude of the force, 𝑚1 and
𝑚2 are the masses of the two objects, and 𝑑 is
the distance between their centers of mass, and
𝐺 = 6.637 × 10−11 Nm2 kg −2 .
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Video Game Gravity
𝐟 = 𝑚𝐠, where 𝐠 = 9.8 m/sec 2 .
This is not physically accurate, but then again,
neither is being able to jump two or three times
your own height, steer in midair, or double
jump. When it comes to jumping in video
games, reality is not just overrated, it's
completely ignored. It just doesn't feel right.
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Video Game Gravity
• In most first-person shooters, when you jump,
you are given an initial burst of upward velocity,
and then your position is simulated just like every
other airborne object in the world.
• In most third-person games your character will
spring up almost instantaneously and reach a
maximum height very quickly. In many games the
character will hover there, then slam back down
on the ground as quickly as it rose up, perhaps
leaving a crater behind.
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Video Game Gravity
• Simulating a jump mechanic using a value of 9.8 m/sec 2
may be even worse, because most players expect a jump to
take a certain amount of time but also expect to be capable
of jumping to unrealistic heights.
• When real-world gravity is used to attain these heights, the
player is in the air too long, and it feels “floaty”.
• Many arcade racing games increase gravity to get the car
back on the ground quickly. The player wants to be in full
control again as quickly as possible, and waiting for realworld gravity to get them back down usually takes too long.
• There are other racing games that use a value of gravity
that is less than the real world value, to facilitate unrealistic
jumps at realistic vehicle speeds.
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Video Game Gravity
• There are also reasons to fiddle with gravity for nonplayer-character objects as well. Sometimes real-world
gravity can create an “objects made of styrofoam”
feeling, so gravity is increased to get an object to tip
over and come to rest more quickly.
• In other situations, an artificially low value of gravity
can make a large object seem even more massive
(especially when accompanied by the right sound
effects), because acceleration on Earth is constant and
is one of a few cues humans instinctively use to
establish an absolute scale for objects in the distance.
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Realism versus Entertainment
• What “feels right" is a subjective matter. It is based
more on player expectation than physical reality.
• In the end, what matters most in a video game is not
what's going on in the CPU or even on the screen, but
what is going on in the player's mind. The human mind
is highly susceptible to suggestion.
• The quest for realism should never be an end unto
itself. A successful video game will harness realism only
where it serves the ultimate goal, which is
entertainment.
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Friction
• The standard dry friction
model is sometimes
called Coulomb friction.
• Charles-Augustin de
Coulomb (1736-1806).
• (Image from Wikimedia
Commons.)
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Static Friction
• When an object is at rest on top of another
object, a certain amount of force is required to
get it unstuck and set it in motion.
• If any less force is applied, the force of friction
will push back with a counteracting force up to
some maximum amount.
• This is called static friction.
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Static Friction
• The following equation is a good
approximation for the maximum magnitude of
static friction: 𝑓𝑠 = 𝜇𝑠 𝑛.
• 𝜇𝑠 is a constant called the coefficient of static
friction that depends on the type of surfaces
rubbing together. Just look it up in a table.
• 𝑛 is the magnitude of the normal force.
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The Normal Force
• The normal force is the force acting perpendicular
to the surfaces that prevent them from
overlapping.
• For example, when an object (such as a bowl of
petunias) is resting on top of another object (such
as a table), the normal force is the force required
to counteract gravity.
• It is the force required to counteract the
component of gravity that acts perpendicular to
the surfaces.
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Normal and Lateral Components
• If the table is at an incline, then we can separate
gravity into a normal component and a lateral
component.
• Inside a computer, we describe the orientation of the
table with a normal vector, and use the dot product to
separate gravity into the relative and normal
components.
• Since the bowl and the table do not accelerate relative
to each other, we know that the normal force of the
table pushing against the bowl must be exactly equal to
the normal component of the force of gravity pulling
the bowl towards the table.
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Not Sliding
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Kinetic Friction
• Once static friction is overcome and the object
is moving, friction continues to push against
the relative motion of the two surfaces.
• This is called kinetic friction.
• The magnitude kinetic friction is generally less
than that of static friction.
• It’s computed the same way of static friction:
𝑓𝑘 = 𝜇𝑘 𝑛.
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Coulomb’s Law of Friction
• The direction of the force of kinetic friction is always
opposed to the relative motion of the surfaces.
• As we said earlier, the coefficient of kinetic friction is
usually less than the coefficient of static friction.
• Thus, if we increase the angle of the table slowly so
that static friction is just overcome, the petunias will
begin to accelerate.
• Coulomb's primary contribution to the theory,
sometimes called Coulomb's law of friction, was that
the force of kinetic friction does not depend on the
relative velocities of the surfaces.
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Spring Forces
• Even if you don't see very many actual
springs in a video game, there are likely
very many virtual springs at work.
• Springs exhibit a general behavior that is
very useful for enforcing constraints, for
example, preventing objects from
overlapping, cloth rendering, and rag-doll
character animation.
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Control Systems
• There are two types of spring motion, damped
oscillation and undamped oscillation.
• A virtual spring (often in the form of a springdamper system) is a type of control system.
• There are certain advantages to be had when the
physical nature of the problem is dropped and we
think of it purely in mathematical terms.
• Indeed, many times the problem was never really
physical to begin with, and was only recast in
physical terms so that the spring-damper
apparatus could be applied.
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The Rest Length
• Consider a spring with one end fixed and the
other end free to move in one dimension.
• When the spring is at equilibrium with no
external forces on it, it has a natural length,
called the rest length.
• If we stretch the spring, then it will pull back
to try to regain its rest length.
• Likewise, if we compress the spring, it will
push back.
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Rest length
Compress
Stretch
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Hooke’s Law
• Robert Hooke (1635 –1703).
• (Image from Wikimedia
Commons.)
• The magnitude of the
restorative force is
proportional to the distance
from the rest length
(provided the force does not
exceed the elastic limit of the
material used to construct
the spring).
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Hooke’s Law
𝑓𝑟 = 𝑘(𝑙𝑟𝑒𝑠𝑡 − 𝑙)
𝑙𝑟𝑒𝑠𝑡
Where 𝑘 is the spring constant
that describes how stiff the
spring is, 𝑙𝑟𝑒𝑠𝑡 is the spring’s rest
length, and 𝑙 is the length that
the spring has been stretched or
compressed to.
𝑙
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Rewriting Hooke’s Law
• Things get easier if we adopt a reference
frame where the position x = 0 designates the
rest position, in which the spring has its rest
length and there are no restorative forces.
• Let 𝐾 = 𝑘 𝑚 . Since 𝐾 contains both the
spring constant 𝑘 and the mass of the particle
𝑚, it measures the spring's ability to
accelerate a particle at its end.
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Rewriting Hooke’s Law
• With those notational changes, we can rewrite
Hooke’s Law as 𝑎 𝑡 = −𝐾𝑥(𝑡), where 𝑎 𝑡 is
acceleration as a function of time and 𝑥 𝑡 is
position as a function of time.
• This is called a differential equation, since it is
an equation in both position 𝑥 𝑡 and its
second derivative, acceleration 𝑎 𝑡 .
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Solving Differential Equations
• We don’t have the tools to
solve general differential
equations, but this one is
not too hard.
• If we grab a spring and
experimentally graph the
position of its end as a
function of time after
compression, we get a
graph like this:
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Solving Our Differential Equation
• This function ought to look familiar to you: it's
the graph of the cosine function.
• Let's see what happens if we just try 𝑥(𝑡) =
cos(𝑡) as our position function. Differentiating
twice to get the velocity and acceleration
functions, we get:
𝑥(𝑡) = cos(𝑡)
𝑥 𝑡 = − sin(𝑡)
𝑥 𝑡 = − cos(𝑡)
which is close, but we're missing the factor of 𝐾.
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Does 𝐾 Matter?
• To understand where 𝐾 should appear in 𝑥(𝑡),
consider what happens to the graph of 𝑥(𝑡)
when we change the value of 𝐾.
• In other words, we repeat our physical
experiment and vary the stiffness of the spring.
• The result is that larger values of 𝐾 (stiffer
springs) result in a graph that is horizontally
compressed: the frequency of oscillation is
increased.
• Likewise, smaller values of K cause the spring to
oscillate more slowly, and the graph is expanded.
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How Much Does 𝐾 Matter?
• Furthermore, we observe that the frequency is
proportional to the square root of 𝐾. For
example, when we increase 𝐾 by a factor of four,
the frequency doubles.
• This gives us a hint as to where 𝐾 should appear,
since all we are doing is scaling the time axis.
𝑥(𝑡) = cos( 𝐾𝑡)
𝑥 𝑡 = − 𝐾 sin( 𝐾𝑡)
𝑥 𝑡 = −𝐾 cos( 𝐾𝑡)
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Angular Frequency
The quantity 𝐾 is called the angular frequency
and comes up often enough that we find it
helpful to introduce the notation
𝜔 = 𝐾= 𝑘 𝑚
and we can write the solution as
𝑥(𝑡) = cos(𝜔𝑡)
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Three More Degrees of Freedom
There are some degrees of freedom inherent in the
motion of the spring that we have not accounted for.
1. We are not accounting for the maximum
displacement, known as the amplitude of the
oscillations and denoted 𝐴. Our equation always has
an amplitude of 1.
2. We are assuming that 𝑥(0) = 𝐴, meaning the spring
was initially stretched to the maximum displacement
𝐴 and released with zero initial velocity. However, in
general, we could have pulled it to displacement 𝑥0 ≠
𝐴 and then given it a shove so it has initial velocity 𝑣0 .
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The Three are Two
• It would appear that we have three more variables that
need to be accounted for in our equation if it is going
to be completely general.
• As it turns out, the three variables we have just
discussed (the amplitude, initial position, and initial
velocity) are interrelated.
• If we pick any two, the value for the third is fixed.
• We'll keep 𝐴 as is, but we'll replace 𝑥0 and 𝑣0 with the
phase offset 𝜃0 , which describes where in the cycle the
spring is at 𝑡 = 0.
• Adjustments to the phase offset have the simple effect
of shifting the graph horizontally on the time axis.
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Simple Harmonic Motion
Adding these two variables, we arrive at the
general solution, the equations of simple
harmonic oscillation:
𝑥(𝑡) = 𝐴 cos(𝜔𝑡 + 𝜃0 )
𝑥 𝑡 = −𝐴𝜔 sin(𝜔𝑡 + 𝜃0 )
𝑥 𝑡 = −𝐴𝜔2 cos(𝜔𝑡 + 𝜃0 )
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Damping Forces
• So far, we have been studying a physically nonexistent
situation in which the spring will oscillate forever.
• In reality, there are usually at least two more interesting
forces, driving force and friction.
• The driving force is an external force, that acts as the input
to the system and causes the motion to begin. Friction we
have already met.
• The general term used to describe any effect that tends to
reduce the amplitude of an oscillatory system is damping,
and we call oscillation where the amplitude decays over
time damped oscillation.
• Damping forces are useful in video games, so let's discuss
them in more detail.
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Damping Force
The damping force is simply 𝑓𝑑 = −𝑐 𝑥, where 𝑓𝑑
indicates the instantaneous magnitude and
direction of the damping force, 𝑥 is the
instantaneous velocity, and 𝑐 is a constant that
describes physical conditions such as the
viscosity, roughness, etc.
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Qualitative Observations
• The damping force has an extremely simple form, but things get
interesting when we study motion over time.
• Qualitatively, we can make some basic predictions about how
damped oscillation of a spring would differ from undamped
oscillation of the same spring.
• The more obvious prediction is that we would expect the amplitude
of oscillation to decay over time. Like the force of friction, damping
removes energy from the system.
• The second observation is only slightly less obvious: Since damping
in general slows the velocity of the mass on the end of the spring,
we would expect the frequency of oscillation to be reduced
compared to undamped oscillation.
• Those two intuitive predictions turn out to be correct, although, of
course, to be more specific we will need to analyze the math.
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Harmonic Motion with Damping Forces
• Combining the restorative and damping
forces, the net force can be written as
𝑓𝑛𝑒𝑡 = 𝑓𝑟 + 𝑓𝑑 = −𝑘𝑥 − 𝑐 𝑥.
• To derive the equation of motion, we will
need accelerations, not forces. Applying
Newton's Second Law and dividing both sides
by the mass, we have:
𝑓𝑛𝑒𝑡
𝑘𝑥 𝑐𝑥
𝑥 = 𝑚 = − 𝑚 −𝑚
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Spring-Damper Systems in Video Games
Spring-damper systems are used in video games as control
systems. A control system takes as input a function of time
that represents some target value. For example:
1. Camera code might compute a desired camera position
based on the player's position each frame;
2. AI code might determine an exact targeting angle for an
enemy;
3. We may have a desired player character velocity based on
the instantaneous amount of control stick deflection;
4. We might have a desired screen-space position for some
highlight effect, based on the currently selected choice in
a menu.
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The Set Point
• The current value of the input signal is known
as the set point in control system terminology.
• The set point is essentially the rest position of
the spring, and the input signal is like
somebody taking the other end of the spring
and yanking it around.
• It is similar to a driving force, but we are given
a function describing a position rather than a
force or acceleration.
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What Does a Control System Do?
The job of any control system is to take this
input signal and produce an output signal. Using
our examples from 2 slides ago, the output
signal might be (respectively):
1. The camera position to use for each frame
2. The animated targeting angle the enemy will
use to aim the weapon,
3. The player character velocity.
4. The screen-space position of the highlight.
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No Jerks Allowed
• For many control systems, the actual position and set
point are not used; rather, only the error is needed.
• Of course, an obvious question is, if we know the
desired value, why don't we just use that directly?
Because it's too jerky.
• In the same way that the shocks and springs on a car
don't just pass along the elevation of the road directly
to the car, a control system in a video game is often
designed to smooth out the bumps caused by sudden
state changes that might make the camera snap to a
new position or the player jerk into motion.
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PD Controllers
• The camera or screen-space highlight are nonphysical
examples in which the quantity of mass is not really
appropriate and is dropped.
• But the differential equations are still the same, and they
have the same solution.
• Stripped of the spring metaphor, we are left with what is
known as a PD controller.
• The P stands for proportional, and this is the spring part of
the controller, since it is proportional to the current error.
• The damper is the D part, which stands for derivative,
because the action of the damper at any given instant is
proportional to the derivative (the velocity).
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Don’t Reinvent the Wheel (Or the Spring)
• PD controllers (and their more robust cousin,
the PID controller, where the I stands for
integral and is used to remove steady-state
error) are broadly applicable tools.
• They have been standard engineering tools for
decades (centuries?) and are well understood.
• Nevertheless, they are one of the most
frequently reinvented wheels in video game
programming.
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Tuning
• Different cars have suspensions that are tuned differently; sports
cars are tighter and the cars retirees like to drive are smoother. In
the same way, we tune our control systems to get the response we
like.
• Notice that the code uses the 𝑘 and 𝑐 from our earlier equations.
However, most people don't find those to be the most intuitive
dials to have for tweaking.
• Instead, the damping ratio and frequency of oscillation are used for
the designer interface, while 𝑘 and 𝑐 are computed as derived
quantities.
• To tune the frequency, we might adjust either the damped or
undamped version, using either angular frequency or simply Hertz;
the units and absolute value are often not important because the
value that feels good will be determined experimentally anyway.
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Alternatives and Generalizations
• The second-order systems we have described here are certainly not
the only type of control system, nor even the simplest, but they do
behave nicely under a very broad set of circumstances and are easy
to implement and tune.
• Another commonly used control system is a simple first order lag,
𝑥 = 𝑘𝑥, under which the error decays exponentially.
• This is similar to a critically damped second-order system, but with
a bit jerkier response to a sudden change in the set point.
• Another common technique is to chase the set point at a fixed
velocity.
• A filter is another broad class of control system in which the output
is computed by taking some linear combination of set points or
values on previous frames.
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Section 12.3:
Momentum
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Moe’s Box
• Suppose Moe's box has
mass 𝑚, and at a certain
instant we observe it
moving with a velocity 𝑣.
• Coming in late to the
story, we cannot tell what
magnitude of forces were
used to achieve that
motion, or how long the
forces were applied, or
what the history of the
box's velocity was.
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Section 12.4:
Impulsive Forces and Collisions
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Section 12.5:
Rotational Dynamics
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Section 12.6:
Real-Time Rigid Body Simulators
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That concludes Chapter 12. Next, Chapter 13:
Curves in 3D
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