Chapter 9. triangles, trig and trajectories
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castle defense needed
Camelot - we have a problem!
The Head First castle is in imminent danger! Back when it was
built, the longest ladder available from Sieges-R-Us was 15.0 m
long. So the castle was designed with a moat 15.0 m wide and
a wall 15.0 m high, making it impossible for anyone to put a
ladder from the edge of the moat to the top of the wall.
End of ladder
is nowhere near
top of wall.
Ladder
15.0 m
Moat filled
with water
Bottom of
ladder is at
edge of moat.
15.0 m
Wall
15.0 m
But the Sieges-R-Us website has just been
updated with a new top of the range 25.0 m
ladder. It’s only a matter of time before someone
comes to attack your castle armed with the new
ladder, and your current defense system just isn’t
big enough...
New ladder will
comfortably reach
top of wall!
Wall
25.0 m
Ladder
Moat
15.0 m
15.0 m
The new ladder
If you don’t act quickly, someone will turn up
with the new ladder, and you’ll be toast. It’s time
to design a new castle defense system!
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triangles, trig, and trajectories
Check the box
next to the idea
you think is best.
Here are some ideas for a new castle defense system.
Unfortunately, you don’t have many spare stones lying around, only
some shovels and food.
Write down at least one advantage and disadvantage of each idea,
and check the box next to the idea you think is the best.
Coat the top of the wall with something slippery.
Advantage(s)
Disadvantage(s)
Make the moat wider.
Advantage(s)
Disadvantage(s)
Run away! Run away!
Advantage(s)
Disadvantage(s)
Make the wall higher.
Advantage(s)
Disadvantage(s)
Put a health and safety rep on the top of the wall to recite the
working height directive repeatedly to anyone who gets higher
than 2 m. Also to ask when their ladders were last inspected
and if they have been trained in the proper use of ladders.
And insist that they all wear safety harnesses, hard hats, ear
defenders, goggles, gloves and toe protectors.
Advantage(s)
Disadvantage(s)
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we need a wider moat
Here are some ideas for a new castle defense system.
Unfortunately, you don’t have many spare stones lying around, only
shovels and food.
Write down at least one advantage and disadvantage of each idea,
and check the box next to the idea you think is the best.
Coat the top of the wall with something slippery.
Advantage(s)
Ladders will slip, it might
delay them a bit.
Disadvantage(s)
Washes off in the rain, attracts
cats, not a long-term solution.
Make the moat wider.
Disadvantage(s)
Advantage(s)
I’m not sure how wide to make
the moat.
I have a shovel to dig a
wider moat.
Run away! Run away!
Advantage(s)
I won’t be there when they
get into the castle.
Disadvantage(s)
They’ll get into the castle.
Make the wall higher.
Advantage(s)
This would definitely keep
them out of the castle ...
Disadvantage(s)
... but there’s no stone available
to build a higher wall.
Put a health and safety rep on the top of the wall to recite the
working height directive repeatedly to anyone who gets higher
than 2 m. Also to ask when their ladders were last inspected
and if they have been trained in the proper use of ladders. And
to Insist that they all wear safety harnesses, hard hats, ear
defenders, goggles, gloves and toe protectors.
Advantage(s)
The attackers might get bored
and attack a different castle.
Disadvantage(s)
By wearing ear defenders, they
can ignore him and continue
attacking the castle anyway.
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It’s a lot easier to
dig a wider moat
than it is to build
a higher wall.
triangles, trig, and trajectories
How wide should you make the moat?
Widening the
moat moves the
bottom of the
ladder further
away from the
top of the wall.
The best way of defending the castle against the
25.0 m ladder is to make the moat wider. The
moat is already 15.0 m wide—so how much
wider do you need to make it?
You could try making the moat the same size as
the ladder —25.0 m— so that the distance from
the edge of the moat to the bottom of the wall is
the same length as the ladder. That would make
sure that attackers couldn’t simultaneously put
one end of the ladder at the edge of the moat
and the other end on the top of the wall.
You want the distance from
the edge of the moat to
the top of the wall to be
more than 25.0 m.
15.0 m
? m
Old moat width
But time is of the essence, and you don’t want to start out
digging a 25.0 m moat if a narrower moat will do the same
job. The important thing is the distance from the edge
of the moat to the top of the wall. If that’s more than
25.0 m, there won’t be anything to lean the ladder on. And
you might be able to achieve that with a narrower moat... a
sketch should help.
a. Draw a sketch of the 15.0 m castle
wall, 25.0 m ladder, and extended
moat, where the ladder is only just
too short to reach the top of the
wall from the side of the moat.
(This sketch is just a quick drawing
to get the visual parts of your brain
working—the lengths on it don’t
have to be accurate as long as
everything’s labelled correctly.)
b. What shape does your sketch resemble?
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remember triangles?
Top of ladder doesn’t
quite reach top of wall.
a. Draw a sketch of the 15.0 m castle
wall, 25.0 m ladder, and extended
moat, where the ladder is only just
too short to reach the top of the wall
from the side of the moat.
Ladder
0 m
Minimum width of moat = ? m
This distance is less than
25.0 m. So this way is quicker
and more efficient than
making the moat 25.0 m wide.
b. What shape does your sketch resemble?
It looks like a triangle.
You can turn your complicated-looking castle, ladder,
and moat sketch into a more simple picture of a
right‑angled triangle, with a 90° angle (a right
angle) between the wall and the moat. You already
know the lengths of two of the triangle’s sides and
want to find out how long the third side is.
Simplified version
of our sketch
Ladder
25.0 m
15.0 m
You could figure this out by ordering a 25.0 m
ladder, putting one end at the top of the wall, and
seeing where the ladder touches the ground. But the
attackers might arrive with their new ladder first!
If you don’t want to wait, you can measure 25.0 m
of rope, tie one end to the top of the wall, and see
where it touches the ground when you pull it tight.
But that still involves a lot of steps and equipment.
15.0 m
Moat
Bottom of ladder is
right at edge of moat.
Looks like a triangle, yeah?
Wall
25.
(This sketch is just a quick drawing
to get the visual parts of your brain
working—the lengths on it don’t have
to be accurate as long as everything’s
labelled correctly.)
Wall
Moat
? m
You need to
work out how
long this side is.
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Dawr a little square inat
the corner to show th E.
this is a RIGHT ANGL
triangles, trig, and trajectories
Wouldn't it be dreamy if you
could just work out the smallest moat
width from the triangle drawing, without having
to climb the wall and make a lot of measurements?
But I know it's just a fantasy…
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scale drawings
A scale drawing can solve problems
This is one way of getting
the best moat width from
a triangle drawing. But it’s
not the best way.
At the moment, your drawing is only a sketch—the
triangle’s side lengths aren’t to scale (though the
lengths you’ve written beside them are correct).
A scale drawing is one where you say something like
“1 cm on the drawing = 1 m in real life.” You can then
do the same as you would with the 15.0 m castle wall
and 25.0 m ladder, except with 15 cm and 25 cm!
Start off by making a right-angle to represent where the wall
meets the moat. Then measure up 15 cm to represent the wall.
Now, you want to swing the ladder down and see where it hits
the ground. So set a pair of compasses to 25 cm...
A scale drawing takes
time, and I’m sure someone
must have solved a problem
like this before. Is there an
equation that’ll help?
m
25 cwn).
o
t
s
o
passeladder d
m
o
c
of
he
pair winging t
a
t
Se e as s
(sam
Make a right-angle here.
Measure up
15 cm for
the wall.
... and swing down from the top of the wall to meet the ground
line. Now you can measure the most economical moat width.
That’s a lot of effort just to work out a simple length though ...
Sweep compasses
around until you cross
the horizontal line.
You can solve some problems
with scale drawings, but it
takes time and effort.
342 Chapter 9
25
cm
15 cm
? cm
Accurate scale version
of your triangle.
Measure this side and
scale it up to get the
best moat width.
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triangles, trig, and trajectories
Pythagoras’ Theorem lets you figure out the sides quickly
Pythagoras’ theorem is an equation for solving this
kind of problem without waving ladders around or
making a super-accurate drawing. You only need to
know the lengths of two sides, and the equation will tell
you the third.
Longest side
is called the
hypotenuse.
The longest side of the triangle is opposite the biggest
angle (the right angle). This side has a special name and
gets called the hypotenuse.
The hypotenuse is
opposite the right angle
- the largest angle.
If you square the length of the hypotenuse, the answer is
equal to the answer you get if you square the length of the
other two sides individually, then add the squares together.
c
That’s very wordy. So here’s the equation - if you label the
sides of your triangle a, b and c (where c is the hypotenuse)
then Pythagoras’ Theorem says:
c2 = a2 + b2
b
It doesn’t really matter which
letters you use for the sides.
We’ve chosen the same letters as
the AP physics equation table.
a
c2 = a2 + b2
The hypotenuse is on one side of
the equation, the rest are on the
other side of the equation.
Pythagoras’ Theorem only works for right‑angled
triangles. You can’t use it if your triangle doesn’t have a
right angle.
If you already
know two sides of
the right‑angled
triangle, Pythagoras
gets you the third.
Pythagoras only works for
right‑angled triangles.
If the triangle is right‑angled:
The square of the hypotenuse
is equal to the sum
of the other two sides squared:
c2 = a2 + b2
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look for right angles
Q:
You’ve called Pythagoras a theorem and an equation
so far. And I’ve seen things like that called a formula too. So
which is it?
A:
Equation, formula, and theorem mean the same thing really.
They all describe relationships where you write down “something =
another thing.”
Q:
Q:
What if I know the hypotenuse and want to calculate the
length of one of the other sides?
A:
Q:
You can rearrange the equation so that the side you don’t
know is on its own on the left.
So where does Pythagoras’ Theorem come from? Aren’t
we going to go through proving it?
How do I try to remember Pythagoras? I mean, how do
I remember which sides are a, b and c, then what order to put
them in the equation?
A:
If you can remember the form of the equation, you don’t
need to remember the letters. The hypotenuse is the longest side,
whatever letter you use to name it. So hypotenuse goes on the left
of the equation.
Being able to prove Pythagoras’ Theorem doesn’t help with this, so
we’ve not gone into that here.
A:
It’s only really worth going into understanding where an
equation comes from if the understanding you gain helps you see
how the world works, so you can solve physics problems (and other
problems) better.
And on the right of the equation, you square each of the other sides,
then add them together. You can think about the S in SUCK - size
matters. So it’s the square of the longest side that goes on its own.
We’ll probably see a lot of
right‑angled triangles in physics
because the ground’s horizontal and
gravity acts vertically?
You’ll spot many right angles between the
horizontal and vertical
Right‑angled triangles are going to be one of your most
important physics tools. There are lots of right angles in
physics, often between the horizontal ground and vertical
walls or vertical acceleration vectors that exist as a result
of gravity.
Keep your eyes open for them as this chapter progresses ...
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triangles, trig, and trajectories
Sketch + shape + equation = Problem solved!
Back to the castle and the new Sieges-R-Us ladder!
You started with a sketch and spotted a right‑angled triangle shape in it. After toying
with the idea of a scale drawing, Pythagoras popped up with an equation!
So now you can work out the best moat width—and save the castle!
c2 = a 2 + b 2
Ladder
Wall
0 m
25.
15.0 m
25.0 m
15.0 m
c
b
Moat
Minimum width of moat = ? m
? m
Start with a sketch
Look for familiar shapes
(triangles, rectangles, etc)
a
Use an equation that tells you
about this kind of shape
Solve your problem!
A castle is built on flat ground with 15.00 m walls. How wide must the moat be to
ensure that a 25.00 m ladder only just touches the top of the wall? Assume that the
base of the ladder is placed at the edge of the moat.
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pythagorean theorem
A castle is built on flat ground with 15.00 m walls. How
wide must the moat be to ensure that a 25.00 m ladder
only just touches the top of the wall? Assume that the
base of the ladder is placed at the edge of the moat.
Start with a sketch:
Ladder
m
5.0
Wall
15.0 m
2
or 32 = 3×3 = 9.
If you take the square root (√ ) of a number, the
answer you get is the number you’d have to
square to get the one you started off with. For
example, √9 = 3 because 32 = 9.
Want to know a, the width of the moat.
By Pythagoras, c = a + b
Rearrange
equation
for the side
you want.
A:
?m
2
There were square roots in that solution,
but it’s been a while since I used these, and
they’re a bit hazy. Remind me how they work
again?
As we saw in chapter 3, the square of a
number is the number times itself.
Moat
This is what you say
if you use Pythagoras.
Q:
2
2
a2 = c2 - b2
Q:
I noticed that I got nice round numbers –
15.0, 20.0 and 25.0 – for the side lengths. Does
that always happen with right‑angled
triangles?
A:
a2 = 252 - 152
a = 400
2
a = 400 = 20.0 m
So the best moat width is 20.0 m (3 sd).
Values in your question
were given to 3 sd, so
your answer should
have 3 sd too.
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Here, the wall, ladder and moat formed a
right‑angled triangle which has nice side lengths
in a 3:4:5 ratio. 32 + 42 = 52, and a 15:20:25 ratio
is just a 3:4:5 ratio multiplied by 5.
But usually that doesn’t happen - your calculator
will give you answers that you’ll have to round
to the same number of significant digits as the
values you were initially given to work with.