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Chapter 12. using forces, momentum, friction and impulse

# 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!

phics in place, but need

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.

472   Chapter 12

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

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!

you are here 4   473

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!

a sketch to work out what physics

the story involves. What’s it LIKE?

474   Chapter 12

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

handle passing using

equations of motion!

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

you are here 4   475

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

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,

Do they become one object?

476   Chapter 12

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

that stopped him completely? Where’s

the momentum conservation there?

A:

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

m1

p1

p2

m2

But sometimes they run in

at different angles before

the tackle happens.

m1

p1

m2

p2

you are here 4   477

How are we gonna to figure out what

happens if they hit each other at an angle

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

You can’t use

what you know of

Pythagoras, sine,

cosine or tangent

is right-angled.

p2

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 ..

478   Chapter 12

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.

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…

you are here 4   479

right-angled triangles

Use component vectors to create

some right-angled triangles

1. You need to add together vectors at an angle.

p1

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

480   Chapter 12

gets you to the same

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?

you are here 4   481

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.

482   Chapter 12 