Chapter 12. using forces, momentum, friction and impulse
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simfootball
It’s ... SimFootball!
You’ve been contacted by the SimFootball team, who
need your help with some of the physics in their
video game. If you can help them figure out why the
characters in the game aren’t behaving like they would
in real life - you’ll get an all expenses paid trip to the
X-Force Games.!
Memo
From: SimFootball
Re: Physics in our new
game
We saw you on FakeBu
sters the other night, and
thought you might like to
be a consultant on our
latest game.
You can help, right?
That trip to the X-Force
Games will be sweet! I
need a vacation!
We already have the gra
phics in place, but need
advice on the physics en
gine for many of the
components of the game
- passing, tackling, tire
drag (in training mode)
and kicking. You will
work closely with one of
our programming team.
If you can help us get thi
s all together in time,
we’ll send you to the X-F
orce Games...all
expenses paid.
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using forces, momentum, friction and impulse
The SimFootball programming team have come up with a list of things they need physics advice on for
their game. Your first job is to outline the physics you think you’ll need to use.
So start with a sketch of each item to reduce it to its ‘bare bones’ and see if it’s like something you
already know how to do. Label things like velocity, acceleration, force etc where appropriate. And give a
brief outline of the kind of physics you might use to solve each problem.
a. Passing - Working out the path of a ball that
has been thrown through the air at a known angle
with a known initial velocity.
b. Tackling - Players with known masses each
running with a certain velocity collide with each
other and grab on.
c. Tire drag - In training mode, a player with a
rope around his waist runs, dragging a tire along
the ground.
d. Kicking - Moving foot kicks stationary ball with
a force, and is in contact for a known period of
time.
Don’t worry if you don’t know
much about football.
Each of the game elements are explained in the
‘Sharpen your pencil’. You’re only going to be
working with the physics, so it doesn’t matter if
you don’t know much about the rules.
This is American
football, not soccer.
But whatever you
call it, don’t worry!
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start with a sketch
The SimFootball programming team have come up with a list of things they need physics advice on for
their game. Your first job is to outline the physics you think you’ll need to use.
So start with a sketch of each item to reduce it to its ‘bare bones’ and see if it’s like something you
already know how to do. Label things like velocity, acceleration, force etc where appropriate. And give a
brief outline of the kind of physics you might use to solve each problem.
a. Passing - Working out the path of a ball that
has been thrown through the air at a known angle
with a known initial velocity.
b. Tackling - Players with known masses each
running with a certain velocity collide with each
other and grab on.
This looks like a
a = -9.8 m/s2
projectile fired
through the air at an
v0
angle. Use equations
v0v
of motion and treat
horizontal and vertical v0h
components separately.
Players both have mass Before
and velocity, so both
m1
m
have momentum before
v1
v2 2
collision.
Momentum is conserved After
m1 m2
so it must be the same
before and after.
v=?
c. Tire drag - In training mode, a player with a
rope around his waist runs, dragging a tire along
the ground with a constant velocity.
d. Kicking - Moving foot kicks stationary ball with a
force, and is in contact for a known period of time.
The tire is being
pulled at an angle,
so you can maybe
make a rightangled triangle
and use component
vectors of forces
to work this out.
Foot and ball both
have a mass and a
velocity, and again
momentum must be
conserved.
m2
m1
Force of ball
on foot.
You don’t know exactly how to
do some of these problems yet,
but don’t worry - you’ve already
got off to a great start!
If you’re given a story, start with
a sketch to work out what physics
the story involves. What’s it LIKE?
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Force of
foot on ball.
t = contact time
If two objects interact, look out
for being able to use momentum
conservation or a form of
Newton’s 2nd Law (either F = ma
or F∆t = ∆p) as both objects
experience the same size of force.
using forces, momentum, friction and impulse
We can already
handle passing using
equations of motion!
Jim: Yeah, but what about tackling? The players usually hit head
on and grab on to each other. In the game we know their masses
and velocities before the tackle. Ow!!! What are you doing?!
Joe: Just being a part of it! Looks like if I’m running faster when I
tackle you, we move faster afterwards than when I run slowly.
Frank: And if your mass was larger, Jim would have gone flying!
Joe: The total momentum, mass × velocity, will be the same
before and after - right?
Jim: I’m glad we’re back to math now! Yeah, the game would need
to move the players with the correct velocity after the tackle. We
know the mass and velocity of each player before the tackle, so
using momentum conservation sounds about right.
Two football players hit each other head on. One has a mass of 95.0 kg and is running from left to right at
8.50 m/s. The other has a mass of 120.0 kg and is running from right to left at 3.80 m/s
If the players lock together in the tackle, what velocity do they move with in the split second after the tackle?
Hint: If the
players lock
together, they
move as one
mass after
the tackle
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objects and collision
Two football players hit each other head on. One has a mass of 95.0 kg and is running from left to right at
8.50 m/s. The other has a mass of 120.0 kg and is running from right to left at 3.80 m/s
If the players lock together in the tackle, what velocity do they move with in the split second after the tackle?
Before:
m1 = 95.0 kg
v1 = 8.5 m/s
After:
m2 = 120.0 kg
v2 = -3.8 m/s
m3 = 95.0 + 120.0 = 215.0 kg
v3 = ?
Left to right is positive.
Use momentum conservation to work out v3:
total momentum before = total momentum after.
m1v1 + m2v2 = m3v3
mv + m v
v3 = 1 1 m 2 2
3
It’s safest to
rearrange your
equation before you
put the values in.
Momentum is a VECTOR so you
need to choose which DIRECTION
to define as positive.
v3 =
95.0 × 8.50 - 120.0 × 3.80
215.0
v3 = 1.63 m/s (3 sd)
They go from left to right at 1.63 m/s (3 sd).
Momentum of
each player.
Momentum is conserved in a collision
Momentum is always conserved in an interaction between
two or more objects. So when the two players collide in the
tackle, the total momentum must be the same afterwards as it
was before the collision.
This happens because each player experiences the same size
of force when they collide, but in opposite directions - a
Newton’s Third Law pair of forces. The same size of
force always causes the same change in momentum.
So the first object has its momentum changed in the direction
of the force acting on it - and the second object has its
momentum changed in the direction of the force acting on it.
But the forces are equal sizes and in opposite directions. So
the changes in momentum are equal sizes and in opposite
directions. This means that the total momentum is the same
both before and after the collision. The changes in momentum
make no difference to the total when you add them together.
Before
m1
p1
Add momentum
vectors by
lining them up
‘nose-to-tail’.
After
p2
m2
p1
ptot
p2
m1+m2
Players form one
mass after collision.
Total momentum
is conserved.
ptot
When two objects collide,
think about what happens.
Do they become one object?
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using forces, momentum, friction and impulse
Q:
How do you know that the two
masses that exist before the collision
have turned into one mass afterwards?
A:
You’ll often do problems where two
masses stick together after colliding. This
means that they no longer move as two
separate masses, but as one mass with a
single velocity. Read the question carefully!
Q:
Are there any buzzwords that
indicate that the objects stick together?
A:
Sometimes the term “inelastic” is used
to indicate a situation where two objects
collide without bouncing (in an “elastic” way).
Q:
Is momentum always conserved?
Or does that only happen when the
objects stick together?
A:
Momentum is always conserved
in any interaction between two objects,
whether they stick together or bounce off
of each other. This happens because each
object experiences an equal-sized forces in
opposite directions as a result of the collision.
The same size of force always leads to the
same change in momentum.
So if one object’s momentum changes
by +10 kg.m/s and the other object’s by
-10 kg.m/s, the total momentum is still the
same. The +10 and -10 add to zero when
you add the “after” momentums together.
Q:
So that happens because of a
Newton’s Third Law pair of forces?
A:
Spot on! Newton’s 3rd Law and
momentum conservation are two sides of the
same coin.
Q:
What if the football player had a
collision with an advertising billboard
that stopped him completely? Where’s
the momentum conservation there?
A:
The advertising billboard is attached
to the Earth, which has a huge mass
compared to the player. As momentum is
mass × velocity, the Earth’s huge mass
means that the change in its velocity is far
too small for you to notice.
But the collision might be at an angle
The SimFootball team are really happy with what you
told them about tackling, and write it into the game!
But they soon realize that the problem’s more involved
than they first thought. The players don’t always collide
head on - sometimes they hit each other at an angle.
And they don’t know how to deal with that.
What you did is working out
great ... but the players don’t
always hit each other head on.
Players sometimes hit
each other head-on.
m1
p1
p2
m2
But sometimes they run in
at different angles before
the tackle happens.
m1
p1
m2
p2
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think about angles
How are we gonna to figure out what
happens if they hit each other at an angle
instead of head on like before?
Jim: Well, isn’t momentum still conserved? We can figure out
the total momentum before the collision just like we did before.
This’ll be the the same as the size and direction of the players’ total
momentum after the tackle, when they stick together.
Joe: We can do that in principle ... but in practice it’s going to
be difficult dealing with the momentum vectors if we add them
together to work out the total momentum at the start. Look:
Momentum vectors
for the two players.
p1
When you draw
a sketch, make
sure you think
about angles.
You can’t use
what you know of
Pythagoras, sine,
cosine or tangent
unless your triangle
is right-angled.
p2
Momentum vectors added ‘nose-to-tail’
to work out total momentum.
p1
ptot
p2
Total momentum.
Frank: But what’s the big deal? The vectors make a triangle - and
we can deal with triangles!
Jim: Correction ... we can deal with right-angled triangles. But
that triangle sure ain’t right-angled.
Frank: Oh yeah. When the players hit head on, we didn’t need to
think about angles, because all the action was taking place along a
straight line that ran from left to right.
Jim: But can’t we just use Pythagoras etc?
Joe: Pythagoras only works for right-angled triangles. And what we
know about sine, cosine and tangent only works for right-angled
triangles. I guess we could try to work out something that works for
other triangles, but that sounds waaay hard.
Frank: Hmmm, a triangle with no right-angles like the one we’re
stuck with sure is awkward.
Jim: I wonder if we could somehow flip things around so that there
are some right-angled triangles ..
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using forces, momentum, friction and impulse
A triangle with no right angles is awkward
The main problem with this collision is that the players are running in at
different angles. You can add together the players’ momentum vectors to
get the total momentum before the collision by lining them up nose-to-tail,
like we’ve done here.
Add momentum vectors for
the players ‘nose-to-tail’.
p1
p
Total
momentum
p2
How do you work out the size
and direction of the total
momentum when the triangle
isn’t right-angled?
But the triangle formed by the players’
momentum vectors isn’t right-angled. This
makes it difficult for you to calculate the total
momentum. Pythagoras, sine, cosine and
tangent only work with a right-angled triangle.
A triangle with no right angles is awkward!
Wouldn't it be dreamy if we could
somehow break down that vector triangle
into right-angled triangles that we can work
with. But I know it's just a fantasy…
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right-angled triangles
Use component vectors to create
some right-angled triangles
1. You need to add together vectors at an angle.
p1
If your problem has
two dimensions, think
component vectors.
Vectors aren’t parallel
or perpendicular to
each other.
p2
You can redraw any vector
as two component vectors
at right-angles to each
other. This is especially
useful if you have to add
two vectors together
that aren’t parallel or
perpendicular to each other.
2. Turn each vector into components at right angles.
p1l/r
p1u/d
p1
p2u/d
Total up/down
momentum component
Total left/right
momentum component
pu/d
New component vectors
for the total momentum.
p1l/r
p2l/r
pl/r
Finally, you can make
a new right-angled
triangle out of the
up/down and left/right
components of the total
momentum, and use it
to calculate the total
momentum (which
will be the same before
and after the collision)
p2
Now work with the components! Use
right-angled triangles to add together the
up/down and left/right components of
each momentum vector separately.
3. Add together each set of components.
p1u/d
We’ve used the subscript
‘u/d’ to mean ‘up-down
component’.
p2l/r
p2u/d
You can deal with
right-angled triangles.
We’ve used the subscript
‘l/r’ to mean ‘left-right
component’.
This gives you the up/down and
left/right components of the total
momentum vector.
4. Add new components for total momentum.
p1u/d
pu/d
p1
p
p2u/d
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Adding the components
gets you to the same
place as adding together
the two original vectors.
p2
pl/r
p1l/r
p2l/r
using forces, momentum, friction and impulse
Two players in the “SimFootball” game collide in a tackle and grab on to each other. Their masses and
velocity vectors are shown here:
m1 = 110 kg
a. Calculate the size of the momentum
vector for each player.
29.2°
v1 = 8.86 m/s
m2 = 125 kg
22.4°
b. Draw a sketch to show the left/right and up/down components of each player’s
momentum, and calculate the sizes of these components.
v2 = 2.92 m/s
c. Calculate the size and direction of the total momentum vector using your results from part b.
d. What velocity do the players move with after the tackle?
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what is the momentum?
Two players in the “SimFootball” game collide in a tackle and grab on to each other. Their masses and
velocity vectors are shown here:
m1 = 110 kg
a. Calculate the size of the momentum
vector for each player.
29.2°
p1 = m1v1 = 110 × 8.86 = 975 kg.m/s (3 sd)
p2 = m2v2 = 125 × 2.92 = 365 kg.m/s (3 sd)
v1 = 8.86 m/s
m2 = 125 kg
22.4°
v2 = 2.92 m/s
b. Draw a sketch to show the left/right and up/down components of each player’s
momentum, and calculate the sizes of these components.
p1l/r
29.2°
p1u/d
975
p2u/d
p2l/r 22.4°
365
a
p
a
p
1l/r
cos(29.2) = h = 975
p1l/r = 975 cos(29.2)
p1l/r = 851 kg.m/s (3 sd) right
o
p
o
p
1u/d
sin(29.2) = h = 975
p1u/d = 975 sin(29.2)
p1u/d = 476 kg.m/s (3 sd) down
2u/d
2u/d
sin(22.4) = h = 365
cos(22.4) = h = 365
p2u/d = 365 sin(22.4)
p2l/r = 365 cos(22.4)
p2l/r = 337 kg.m/s (3 sd) left
p2u/d = 139 kg.m/s (3 sd) down
c. Calculate the size and direction of the total momentum vector using your results from part b.
Left/ right components: 851 - 337 = 514 kg.m/s right
Up / down components: 476 + 139 = 615 kg.m/s down
Size: By Pythagoras, p2 = pl/r2 + pu/d2 = 5142 + 6152
pl/r = 514 kg.m/s
θ
pu/d = 615 kg.m/s
p
p = 5142 + 6152 = 802 kg.m/s (3 sd)
Direction: Given angles all measured from the horizontal, so do this too.
o
tan(θ) = a
615 = 50.1° (3 sd) from the horizontal, left to right.
θ = tan-1 514
d. What velocity do the players move with after the tackle?
m = total mass = 110 + 125 = 235 kg
p = mv
v = mp = 802
235 = 3.41 m/s (3 sd) at 50.1° (3 sd) from the horizontal, left to right.
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