Collisions

This section is going to be pretty detailed. It is something that is best read with a physics
textbook handy to get the full effect. It is not required though, and the actual code is
easier to understand than the math involved. Regardless, it is very beneficial to understand
the reason why this works so we can do further research to determine the best way
to model other important physics needed in our games.
Momentum
When the objects stay in motion, it is because of momentum. Objects in motion have
momentum. Momentum is used to measure our object’s mass and velocity. The formula
to calculate momentum is:
p = mv
p is our object’s momentum and we know that m is our object’s mass and v is the velocity.
The reason we know about mass is because of our force formula. We can substitute
our formula for acceleration in our formula for force:
F = ma = m v / t
Now we can multiply our change in time on both sides, which produces:
Ft = ma = m v
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Impulse
Ft is called an impulse. We can do some vector math and multiply our mass by our
change in velocity (the right side of our equation) and see that it can be represented by:
Ft = (mv)
Therefore, we know that our impulse is equal to the change in momentum, which can be
written as follows:
Ft = p
Conservation of Momentum
When objects collide their momentum changes. To be more precise, the magnitude of the
momentum remains the same just in the opposite direction. This is how we can model
our collision response. We can reflect our objects off of each other, knowing that whatever
their momentum was before they collided will remain, but their direction will be
reversed. We just threw two objects into the mix but have only been discussing momentum
on a single object. How does this change our momentum formula? Fortunately, it
does not. This is called the law of conservation of momentum and it means the total
momentum for the objects is constant and does not change. This is true because any
momentum changes are equal in magnitude and opposite in direction. This is expressed
with the following formula:
p1 + p2 = p1 + p2
Kinetic Energy
Now we can discuss Newton’s Third Law of Motion, which basically says that for every
action there is an equal and opposite reaction. Whatever momentum one object
decreases, the other object increases. As momentum is transferred from one object to
another there is another physical property that takes place—kinetic energy. Kinetic energy
is energy associated with moving objects. It is the amount of energy needed to make an
object that is sitting still move. It is also the amount of energy needed to make an object
moving stop and sit still. The formula for kinetic energy is:
Ek = 1⁄2 m v2
When a collision occurs and the amount of kinetic energy is unchanged it is considered
to be an elastic collision. When kinetic energy is lost it is considered to be an inelastic
collision. Objects that collide in the real world will deform and cause a loss of kinetic
energy. If the objects do not deform, no energy is lost.
Coefficient of Restitution
The coefficient of restitution is the measurement of how elastic or inelastic our collision
is based on the types of object that are colliding. The formula for coefficient of restitution
is:
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e = (v2f - v1f) / (v1 - v2)
The coefficient of restitution models the velocity before and after a collision takes place
and the loss of kinetic energy happens. The typical value for e is between 0.0 and 1.0
inclusive. A value of 0.0 means the collision is inelastic and 1.0 means the collision is
elastic. The values in between will have a proportionate elastic collision effect. The
subscripts on the preceding formula are specifying which vectors we are using: 1 and 2 are
the two objects and f is the final velocity of that vector after the impact of the collision.
Conservation of Kinetic Energy
We need to discuss the conservation of kinetic energy, which says that the sum of the
kinetic energy of two objects before they collide will be equal to the sum of the kinetic
energy of the two objects after they collide. The formula for the conservation of kinetic
energy is:
Ek1 + Ek2 = Ek1 + Ek2
Broken down into its components, the formula becomes:
1⁄2 m1 v1
2 + 1⁄2 m2 v2
2 = 1⁄2 m1 v1f
2 + 1⁄2 m2 v2f
2
Solving Our Final Velocities
When we are modeling collisions we need to determine our final velocities (which is what
the f in the earlier formula represents). Before we can do that, we need to expand our
conservation of momentum formula from earlier. We will break down the momentum p
into its components as follows:
(m1 v1) + (m2 v2) = (m1 v1f) + (m2 v2f)
Now, we can solve for our final velocity by combining both of our earlier conservation
formulas with our coefficient of restitution formula. Our final velocities will equate to:
v1f = ( (e + 1) m2 v2 + v1(m1 - e m2) ) / (m1 + m2)
v2f = ( (e + 1) m1 v1 - v2(m1 - e m2) ) / (m1 + m2 )
This uses the conservation of kinetic energy formula with our conservation of momentum
formula, along with our coefficient of restitution, which allows us to solve our final velocity
for each object. That is all we need to start modeling realistic collisions.
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