Chapter 6. Displacement, Velocity, and Acceleration
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what’s going on here?
Just another day in the desert ...
The Dingo pushes the
cage off the platform
as soon as the Emu
rounds the corner.
The Dingo wants the Emu to
stay still for long enough to
deliver an invitation to his
birthday party.
The Dingo needs to
know how high the
platform should be,
and whether the
cage can cope with
falling that far.
The Emu runs at 54
kilometers per hour.
The target is 30 m
from the corner.
The cage will fall
on the target.
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displacement, velocity, and acceleration
... and another Dingo-Emu moment!
Every year it’s the same. The Dingo wants to invite the Emu to his birthday
party - but the daft bird won’t stop running for long enough for him to
deliver the invitation. So this year, the Dingo’s decided that extending a
paw of friendship needs drastic measures. He’s hired a crane, and wants to
push a cage off the platform the moment the Emu rounds the bend. But is
this practical? What height does the platform need to be, and will
the cage be able to handle hitting the ground at a high speed?
Emu - Runningus fasticus
54 kilometers per hour
So the Dingo calls the crane company’s customer service department to ask
some questions ...
Crane Company Magnets
The crane company gets to work on the problem. But we accidentally dropped their memo
and some of the words fell off. Your job is to put them back in the right places. You might
use some magnets more than once, and some not at all.
Also, underline the most important parts in the memo to separate the important stuff
from the fluff - the wheat from the chaff.
To: Dingo
Re: Cage
, tricky! The
g
ding exactly on runnin
lan
e
cag
g
lin
fal
m,
Hmm
- and we
re on the computer up the
is
Emu’s
past the corner. The
get
tar
and
ne
cra
the
up
set
rounds
as the
falls at the same time
the cage falls in the
out the
k
wor
we
If
.
ner
cor
the
to that
,we can set the crane
run
to
Emu
it takes the
. Be careful - the
and take home a fat
less than 25 m/s.
it hits the ground at
is only guaranteed if
30 m
time
Emu
height
54 km/h
cage
commission
distance
speed
54 m/s
velocity
30 km
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magnets solution
Crane Company Magnets - Solution
The crane company gets to work on the problem. But we accidentally dropped their memo
and some of the words fell off. Your job is to put them back in the right places. You might
use some magnets more than once, and some not at all.
Also, underline the most important parts in the memo to separate the important stuff
from the fluff - the wheat from the chaff.
To: Dingo
Re: Cage
, tricky! The
Emu
g
ding exactly on runnin
lan
e
cag
g
lin
fal
m,
Hmm
- and we
54 km/h
re on the computer up the
is
ed
spe
Emu’s
past the corner. The
30 m
get
tar
and
ne
cra
the
up
set
rounds
Emu
as the
falls at the same time
cage
time
the cage falls in the
tance
dis
the
out
the corner. If we work
to that
,we can set the crane
30 m
run
to
Emu
it takes the
. Be careful - the
commission
and take home a fat
height
.
und at less than 25 m/s
teed if it hits the gro
ran
gua
y
onl
is
cage
NOTES
30 km
54 m/s
These didn’t get used
because the units are wrong.
What time does the cage fall for?
What height should the crane be?
Will the cage be going faster than
25 m/s when it hits the ground?
velocity
The Emu’s speed, rather than
his velocity, is important , as
the road is curved.
Which of these would
you try to work out first?
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displacement, velocity, and acceleration
How can you use what you know?
The Dingo drops the cage as soon as the Emu rounds
the corner. Then, the cage falling and the Emu
running both take the same time to reach the target.
The time that the Emu takes to arrive depends on
the speed he runs at and the distance he covers
from the corner to the target. As the Emu always
runs with a constant speed, you already know an
equation you can use to do this.
Once you know the time it takes the Emu to arrive,
you’ll have to figure out how far the cage falls
during that time. This will give the Dingo the height
that he needs to set the platform at.
The cage takes
time to get from
the platform to
the target.
The Emu takes
time to get
from the corner
to the target.
However, if the cage travels faster than 25 m/s in
the time it takes for the Emu to reach the target, this
plan won’t work because the cage will hit the ground
and be destroyed upon impact.
You haven’t dealt with falling
things yet - but don’t worry,
that’s what this chapter’s about!
Hint: You’ll need
to convert units.
These times are equal.
First things first. Work out the time it takes the Emu to cover 30 m from the
corner to the target at a speed of 54 km/h.
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sharpen solution
Work out the time it takes the Emu to cover 30 m from the corner to the target
at a speed of 54 km/h.
Convert units: km/h to m/s
This symbol
means ‘implies 54 km/h in m/s = 54 km x 1000 m
hours
1 km
that’. You
= 15 m/s
can use it
going from
one line to
Work out the time it takes:
the next as
distance
speed =
you rearrange
time
an equation.
speed time =
distance
time =
Rearrange
equation to get
time = ...
distance
speed
=
1 hour x 1 min
x 60
mins
60 s
After stringing together
conversion factors, you’re left
with meters on the top and
seconds on the bottom - m/s.
Equation comes from the units
of speed. Meters per second is
a distance divided by a time.
30 m
= 2.0 seconds (2 sd)
15 m/s
If you don’t feel
so confident about
stringing them together,
you can do the units
conversion one step at a
time. That’s fine too.
The problem gave numbers
with 2 significant digits
to work with, so your
answer should have 2 sd.
The Emu takes 2.0 seconds to
reach the target - so the cage
needs to take 2.0 seconds to
reach the target as well.
NOTES
You know that
the Emu takes
2.0 s to arrive
at the target.
What time does the cage fall for?
The cage falls for 2.0 s.
What height should the crane be?
Will the cage be going faster than
25 m/s when it hits the ground?
00:02.00
So the cage needs to take
2.0 s to fall from the crane.
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00:02.00
displacement, velocity, and acceleration
So I have the time figured out, but uh ... I still don’t know
how high the crane should be, or how fast the cage is going
when it hits the ground. Isn’t that the point?
Don’t be afraid
to start out doing
a question, even
if you’re not quite
sure what direction
it’s going to take.
Don’t worry - you’ve already made progress.
When you started out, you knew a couple of facts about the
Emu’s speed and the distance he covers - but nothing at all
about the cage or the crane platform.
Now we need to figure out how fast the cage is going when
it hits the ground after 2.0 s and the distance it falls in that
time.
BE the cage
Your job is to imagine that you’re the cage.
What do you feel at each of the points
in the picture? Which direction are you
moving in? Are you speeding up
or slowing down? Why are you
moving like this?
At Point 1:
Point 1 - Just
been pushed off
the platform
Point 2
At Point 2:
Point 3
At Point 3:
At Point 4:
Point 4 - Just
about to land
(but hasn’t hit
the ground yet)
Why:
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be the solution
BE the cage - SOLUTION
Your job is to imagine that you’re the cage.
What do you feel at each of the points
in the picture? Which direction are you
moving in? Are you speeding up
or slowing down? Why are you
moving like this??
Point 1 - Just
been pushed off
the platform
Point 2
At Point 1: A ‘special point’, as I’m suddenly going
from standing still to starting to move downwards.
At Point 2: Falling down faster than I was at
point 1.
At Point 3: Falling down even faster than I was at
point 2..
At Point 4: This is the fastest I’ll be going before
I hit the ground (I’ll be here after 2.0 seconds if
the height is right).
Why: Gravity’s accelerating me downwards.
Point 3
Point 4 - Just
about to land
(but hasn’t hit
the ground yet)
In this problem, we gave you headings to use,
but it’s always a good idea to make it clear
which part of the problem you’re answering
at each stage!
The cage accelerates as it falls
We’re going to talk about
the cage’s displacement
and velocity, as the
DIRECTION is starting
to become important
- the cage isn’t being
launched up into the air,
just dropped!
You’ve spotted that the cage accelerates as it falls.
Acceleration is the rate of change of velocity. You
can tell that the cage is accelerating because its velocity
is continually changing. It starts off with zero velocity,
then gets faster and faster until it hits the ground.
With that in mind, it’s on to working out the cage’s
velocity after 2.0 seconds and its displacement in
that time so that the Dingo knows whether the idea’s a
starter - and if so, how high to make the platform.
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You know that
something is
accelerating if
its velocity is
changing.
displacement, velocity, and acceleration
Hey ... what’s with this talk of
displacement and velocity? I was quite
happy with distance and speed.
Displacement and velocity will be
more useful to you in the long term.
As the cage is always falling in the same
direction - straight down - you could use either
distance and speed or displacement and velocity
to describe its motion.
But soon you’re going to be dealing with
situations where direction is crucially important,
and you must use vectors. As you practice using
displacement, velocity, and acceleration for the
cage, you’ll soon get comfortable with them,
which will stand you in good stead in the future.
Displacement, Velocity,
and Acceleration Up Close
Displacement is the ‘vector version’ of
distance and is represented by the letter x in
equations (or the letter s in some courses).
Velocity is rate of change of displacement
- the ‘vector version’ of speed. It is
represented by the letter v in equations.
Acceleration is rate of change of velocity,
represented by a, and doesn’t have a scalar
equivalent. If an object’s velocity is changing,
you need to know which direction the
velocity is changing in for the statement to
have meaning. Otherwise, you don’t know if
the object’s speeding up, slowing down, or
changing direction - which are all ways that
an object’s velocity can change.
‘ Vectorize’ your equation
You’ve already used the equation
distance to work out that it takes
speed =
time
the Emu 2.0 seconds to reach the target.
means
The ‘vector version’ of this equation is
‘change in’
displacement
x
, or v = t.
time
It’s fundamentally the same, except that it
involves velocity and displacement instead of
speed and distance.
velocity =
velocity
v=
x
t
change in
displacement
change in time
We’re using bold letters, like x and v, to represent
vectors and italic letters, like t, to represent scalars.
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displacement or velocity first?
So we need to work out the displacement of the
cage after 2.0 seconds. That doesn’t sound too bad.
Jim: We also need to work out what its velocity will be when it hits
the ground. If that’s more than 25 m/s, then the cage will shatter.
Joe: Why don’t we work out the velocity first? That way, if it turns
out that the cage is going too fast after it’s been falling for
2.0 seconds, we won’t have to bother working out the displacement
as well.
Frank: Sounds good. I’m all for spotting shortcuts!
Jim: Well, we’ve done something similar before with that cyclist
who rode everywhere at the same speed. Can’t we use the equation
x
v = t to work out the cage’s velocity
Frank: Yeah, let’s just use that equation! We want to know the
velocity, and that equation says “v =” on the left hand side. v for
velocity. It’s perfect!
Joe: Um, I’m not so sure. The cage doesn’t have the same velocity
all the time - it accelerates as it falls.
Jim: But we can still use that equation, right? If we work out the
displacement, we can divide it by the time to get the velocity.
Joe: I don’t think so. If the cage always had the same velocity, then,
fair enough, that would work. But the cage’s velocity is always
changing because it’s accelerating - it isn’t constant. We want to
know what its velocity is at the very end, as it hits the ground.
Frank: Oh ... and when it hits the ground, it’s only been traveling at
that velocity for a split second.
Jim: Yeah, as it gets closer to the ground its velocity increases, so
it covers more and more meters per second. If we divided the total
displacement by the total time, we’d get the cage’s average velocity.
Joe: But we need to know what the velocity is the instant it hits the
ground. An average velocity’s no good to us.
Frank: I guess we need to do something different ...
If you calculate the cage’s velocity first, you
won’t have to bother calculating its displacement
if it turns out that the cage will break.
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NEVER blindly stick
numbers into an
equation. Always ask
yourself “What does
this equation MEAN?”
NOTES
What time does the cage fall for?
The cage falls for 2.0 s.
What height should the crane be?
(Come back to this if necessary.)
Will the cage be going faster than
25 m/s when it hits the ground?
DO THIS NEXT!!
displacement, velocity, and acceleration
You want an instantaneous velocity,
not an average velocity
x
The equation v = t works fine if you have
something traveling at a constant velocity. But
the cage gets faster and faster as it falls - and you
want to know what its velocity is the instant it
hits the ground.
∆x
v
∆x
The best you can do with the equation is to work
out the cage’s average velocity, which is the
constant velocity it would need to travel with
to cover that displacement in that time. But since
the cage isn’t traveling with a constant velocity,
this value won’t help you out.
v
∆x
As its velocity increases, the
cage’s displacement is greater
in the same amount of time.
This vector represents the velocity of the
cage just before it hits the ground. The
length of the vector represents the size
of the velocity. Don’t be put off by it
appearing to go ‘into’ the target.
This is the
graph for
the cyclist in
chapter 4.
Plot of displacement vs time
for Alex’s late delivery
x
t
Displacement
(meters)
vavg =
1000
900
800
700
x
600
500
As it falls, the
cage’s velocity
increases.
The acceleration is the
rate at which the velocity
of the cage changes.
v
Strictly speaking, you used distance and
speed rather than displacement and
velocity, but the principle is the same.
x
You’ve previously used the equation v = t to work out the
average velocity of a cyclist who was slowed down by stop lights,
and it gave you the slope of a straight line between the start and
end points of his displacement-time graph. Using the slope of
his displacement-time graph at that point, you were also able to
work out his instantaneous velocity at any point.
400
300
200
How might you try to work out a value
for the instantaneous velocity of the
cage just before it hits the ground.
100
0
0
10
20
30
40
t
50
60
70
80
Time
(minutes)
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methods sometimes work when an equation won’t
So could we draw a displacement-time graph
for a falling thing, and calculate its slope at t = 2.0 s to get
its instantaneous velocity? Will that part still work?
You may be able to use the same method
even if you can’t use the same equation.
As the cage doesn’t fall with a constant velocity, the best you
x
can do with the equation v = t is work out its average
velocity - which isn’t what you want. You can’t reuse this
equation to work out the cage’s instantaneous velocity
because the context is different.
But you can use the same method even if you can’t
directly reuse the same equation. If you draw a
displacement-time graph for a falling thing and are
able to calculate its slope at t = 2.0 s, this will give you
the instantaneous velocity of the cage. As long as you
understand the physics, you can work out how to do a
problem even if you can’t directly use an equation you
already know.
Though you still need to design the experiment...
... but didn’t we already
design an experiment
like this?
Is the experimental setup you now
have in mind similar to what you drew
at the start - or is it different?
If it’s different, draw and label a diagram
of your new experimental setup - and
explain how you’ll use it
to make measurements and draw a
graph that shows you a value for the
displacement at any time.
If it’s the same as what you already
did, you can skip this Sharpen. :-)
Make sure you
include labels so
it’s clear what Electromagnet
everything is.
Ball-bearing
Timer
Clamp stand
Distance from
bottom of
ball-bearing
to top of
switch plate
(tape measure)
Switch plate
Don’t spend too much
time making your
diagram look pretty.
Use the clamp stand and the tape measure to set
the height of the ball-bearing. Time how long
it
takes to fall from that height using the timer,
electromagnet and switch plate. Use a range of
heights, from the smallest the timer can measure
to the height of the ceiling, and several heights
in between as well. And time each height two or
three times to reduce random errors.
The plot a graph with the time along the
horizontal axis and the distance up the vertical
axis. Draw a smooth line through the data points.
The graph lets you read off the time it’ll take
for the ball-bearing to fall any distance.
On your graphs, time should always
be along the horizontal axis.
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Understanding
the physics helps
you to work out
how to solve a
problem even if
you can’t directly
use an equation
you already know.