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Unit 3: Motion
Chapter 8: Average velocity is the rate of change in position
8.1 The Language of Motion
y Magnitude describes the size of a measurement
y The symbol for change= which means delta- Greek letter
y Vector:
- Vector quantities include magnitude and direction
- Vector abbreviations are written in bolded italics with an arrow above
them, such as vfor velocity and dfor position.
- When a direction is written in a vector description, it is usually
abbreviated and put into square brackets, such as 12km [W] for 12 km
west
y Scalar:
- Scalar quantities include magnitude only.
- Scalars are not written with any abbreviations or italics
Measurement Explanation Classification
Distance (d)
(m)
(km)
Shows the extent of a course
between two points
Scalar
Position (d)
(m)
(km)
Shows the location of an
object relative to the point of
origin (where the object first
started off from)
Vector
(includes direction)
Vectors Scalars
Magnitude Magnitude
Direction
Vectors and Scalars
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Time (t)
Seconds (s)
Hours (h)
Describes when an event takes
place
Scalar
Time interval( t)
The difference between initial
time (when an even starts) and
final time (when an event
ends)
*The duration of an event
Scalar
To calculate:
t= tf(final time) ti
(initial time)
Displacement
( d)
(m) (km)
Describes how much an
objects position has changed
in a straight line from the
point of origin.
*if an object ends up back towhere it started from then you
could say that the objects
displacement is zero.
Vector
To calculate:
d= df(final position )
di (initial position)
y It is important to know the different between displacement and distance
- An ant travels 2cm north, 6cm west, 2cm south, and 2cm east.
The ants distance:
2cm+6cm+4cm+2cm= 14cm
The ants displacement:
d= 4cm
The ants displacement is 4cm because it has moved 4cm
from its initial position. It the ant had returned back to its
initial position, then its displacement would be 0m.
Displacement vs. Distance
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y Distance:
W E
A car travelled 7km east to the mall from the gas station. The total distance
travelled from the gas station to the mall is 7 km.
y Position:
If the car traveled 7 km to the mall, what is the cars position? The direction or
position of the car from the gas station is 7 km [E]. If the car was to return back to
the gas station, its position would be 0 km since the car is back to the origin.
y Time interval:
t=0s t=3s t=8s
W E
Suppose a person walking on a sidewalk is walking at 1m/s [E]. She walks from her
house at 0m. She passed the stop sign at 3m [E] and then passed the lamp post at
8m [E]. How long did it take her to walk that distance from the stop sign to the
lamp post?
3m [E] =3s (initial time) 8s 3s = 5s
8m [E] =8s (final time)
Examples
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y Displacement
It took her 5s to walk from the stop sign to the lamp post. Her position at 8s is 8m
[E], and at 3s its 3m [E].
In the time interval of 3s-8s her displacement is
3s=3m [E]
8s=8m [E]
8m [E] 3m [E]
= 5m [E]
Between t=3s and t=5s, the displacement of the girl is 5m [E]. This Vector quantity
represents how much the girls position changed between the time intervals of 3s-
8s. Her displacement from the point of origin (her house) to the lampost is 8m [E].
if she goes back to her house, her displacement would go back to 0m.
y
To indicate opposite directions in vector quantities, opposite signs are used.Here are some common signs that are used in physics to show direction:
North Up +
West East
Left Right
- +
* If an object passes the point of origin, the
direction changes.
South Down -
Direction and Si ns
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y All objects in uniform motion meet the following conditions:
- Travel in equal displacements in equal time intervals
- Objects do not speed up, slow down, or change direction
y
Motion of an object can me analyzed in many ways:- Motion diagram: shows us the objects position at given times and allows
us to visualize the objects motion.
-Position- Time graph: plots position data on the vertical axis (y-axis), and
time data on the horizontal axis. Uniform motion is always represented
as a straight line that runs through the plots on a p-t graph.
*using the plots on a graph to determine the line that
generally fits the plots is called a best-fit line
Uniform Motion and Graphing Uniform Motion
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y The slope of a graph refers to whether a line is horizontal or goes up or down
at an angle. There are 3 types of slopes:
- Positive slope
Line slants up to the right.
Indicates the motion is in the direction of the positive y-axis
Position of the object is increasing in the positive direction from
the point of origin
Since the slope is constant, the motion of the object is
travelling in a uniform motion in the positive direction.
- Zero slopeThe object is not moving
This is also an example of uniform motion, since the displacement
of the ball during any time interval is constant.
- Negative slope
Line slants down to the right
Indicates motion is towards the direction of the negative y-
axis, or returning back or past the point of origin.
*This graph displays the
various slopes on a
position-time graph. For
this chapter refer to the
red line. The blue and
green lines will be
explained further in the
summary.
Slope on a position-Time
Positive slo e
Zero slope
Ne ative slo e
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8.2 Average Velocity
y Speed (v) is the distance an object travels during a time interval divided by
the time interval.- Speed is a scalar quantity.
- The SI unit for speed is metres per second (m/s)
- Ex. A toy car travelled 40m in 10s. What was its speed?
40m/10s= 4m/s
y Velocity (v) is the displacement of an object during a time interval divided by
the time interval except
- Velocity describes how fast an objects position is changing- Velocity is a vector quantity and includes direction
- The SI unit for velocity is metres per second (m/s)
- The direction of velocity is the same as the direction of the
displacement or position
- Ex. A toy car travelled 50m [E] in 5s. What is its velocity?
50m [E]/5s= 10m/s [E]
- Objects travelling the same speed can have different velocities simply
because they might be travelling in different directions.
One can have a positive velocity, while the other speed can
have a negative velocity.
- Velocities change when magnitude or direction or both change.
How to use displacement and time intervals to calculate the rate and
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y The slope of a graph represents
rise/run, which is the change in
the y-axis divided by the change
in the x-axis.
y On a position time graph the
slope is the change in position
( d) divided by the change in
time ( t)or the objects average
velocity-v av
y The steeper the slope, the
greater the change in
displacement during the same
time interval. In other words, the object is changing its position quicker in the
same time period.
y To calculate the slope:
- Slope= d
t= df di (final displacement initial displacement)
tf ti (final time initial time)
= 15m 5m
6s 2s
= 10m
4s
= +2.5 m/s- This is the objects average velocity. Because we have included the
direction (+) it is the velocity since velocity is a vector. If we had not
included the direction, and left it as 2.5 m/s, this would become the
objects speed.
Calculating the slope of the Position-Time graph
Rise
Run
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- Also, because it is a position time graph, it automatically becomes
velocity because position is a vector and has direction, making the
answer to the slope a vector as well.
y The slope of a position-time graph represents the objects average velocity.
- positive slope= average velocity is forward
- negative slope= average velocity is backward
- Zero slope= average velocity is zero.
y The relationship between velocity, displacement and time can be described
by the following diagram:
D= displacement
V= velocity
T= time
y It is important to know the difference between velocity and speed
Average Speed= d Average Velocity= d
t t
Ex. A butterfly flies 5m east, 3m south, and 5m west over a time interval of12s.
5m 3m
5m
- Average speed:
13m/12s= 1.08m/s
- Average velocity:
3m [S]/12s= 0.25m/s [S]
Velocity vs. Speed
D
TV
V= D/T
T=D/V
D=VT
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y To convert between m/s and km/h:
- Multiply m/s by 3.6 to convert to km/h m/s 3.6 km/h
- Divide km/h by 3.6 to convert to
- Ex. 85km/h: divide by 3.6= 23.6m/s
- Ex. 72m/s: multiply by 3.6= 259km/h
y To convert between hours and minutes:
- Multiply hours by 60 to convert to minutes
- Divide minutes by sixty to convert to hours
Chapter 9: Acceleration is the rate of change in velocity.
9.1 Describing Acceleration
y An object travelling with uniform motion has equal displacements in equal
time intervals.
y An object travelling with non-uniform motion will:
- Have different displacements during equal time intervals
- Take different amounts of time to travel equal displacements
- Have a continuously changing velocity
y A change in velocity changes when the speed or direction of motion of the
object changes.
y A change in velocity can be calculated by subtracting the initial velocity from
the final velocity.
v= vf vi
Converting Between m/s and km/h
-
Positive and Negative Changes in Velocity
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y Positive changes in velocity:
- If the change in velocity is in the same sign (+/-) as the initial velocity,
the speed of the object is increasing.
Ex. Tom is roller blading down the
street at +8m/s in a forward direction
(+). He is in a hurry so he increases
his velocity to +12m/s. His change in
velocity is
+12m/s (+8m/s) = +4m/s
*The change in velocity is +4m/s in
the forward direction. This means,
Tom was speeding up by 4m/s in the
original direction. His original
forward direction was positive, so his
change in velocity is also positive when he sped up.
y Negative changes in velocity:
- If the change velocity is in the opposite of the initial velocity, the speed
of the object is decreasing.
Ex. Suppose Tom reaches his destination and slows down from +12m/sto +3m/s forward. His change in velocity is
+3m/s (+12m/s) = -9m/s
*The change in velocity is -9m/s opposite the forward motion. In other
words, Tom was slowing down by 7m/s in the original direction. If the
initial forward direction was positive and the change in velocity is
negative the object is speeding up.
y Constant velocity:- Any object travelling with uniform motion in a straight line would have
zero change in velocity.
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y Acceleration (a) is the rate of change in velocity, or how quickly the velocity
of an object is changing.
- Acceleration can be due to a change in speed and/or a change in
direction.- When talking about acceleration, we need to include the magnitude of
the change in the velocity of the moving object and the direction of the
objects velocity- because acceleration and velocity are vectors.
- When comparing the acceleration of two objects, the object with the
greater acceleration changes its velocity in a shorter time interval and
has a greater change in velocity during the same time interval.
Ex. Lets look at the following situation:2 cars, a race car and a beetle, are having a race. Both cars
will be able to reach the speed of
60km/h; however the race car will
be able to reach it faster because of
its powerful engine. In other words, the race car
will be able to change its velocity faster and
therefore will have a greater acceleration, than
the beetle.
*When an object is accelerating the velocity changes and its motion is not
uniform.
y In a straight line motion, acceleration can be either positive or negative, and
is the same as the direction of the change in velocity.
Acceleration
Positive and Negative Acceleration
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y Positive acceleration:
- A car driving along a straight road changes its velocity from 40km/h to
60km/h. the forward motion of the car can be represented as positive
(+). When the cars speed is increasing, the car has a positive
acceleration.
y Negative acceleration:
- Now the car approaches a red light and needs to slow down. The driver
pressed the brakes and the cars speed begins to decrease in a forward
motion. The car now has a negative acceleration.
Velocity
Acceleration
Velocity
Acceleration
If an object is increasing its speed in apositive direction then change in velocity
is positive therefore the acceleration is
positive also.
If an object is decreasing its speed in a
positive direction then the change in
velocity is negative therefore the
acceleration is negative also.
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Velocity
Acceleration
Velocity
Acceleration
*If an object is changing its velocity in the same direction as its initial position then
both the velocity and acceleration is positive, even if it is slowing down.
*If an object is changing its velocity in a different direction than its initial positionthen both the velocity and accelerations is negative, even if it is speeding up.
9.2 Calculating Acceleration
y Acceleration is measured in units of m/s2 (metres per second squred)
y The motion of an object with a changing velocity can be represented on:
- A velocity-time graph
- A position time graph
If an object is increasing its speed in a
negative direction then the change in
velocity is negative therefore the
acceleration is negative also.
If an object is decreasing its speed in a
negative direction then the change in
velocity is positive therefore theacceleration is positive also.
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y The slope of a velocity-time graph is the objects average acceleration.
- Slope= rise/run: when this is calculated on a v-t graph, the answer is the
objects acceleration
- Shows how fast the object is changing its velocity
Velocity-Time graphs
y When the slope on a velocity-time graph is
touching the x-axis at any point, this means
that the object is stationary.
y Zero slopes on a v-t graph mean that the
objects velocity is constant- the object is still
moving but at a constant speed.
y Positive slopes on a v-t graph mean that the
objects velocity is increasing in a positive
direction.
y Negative slopes on a v-t graph mean that
the objects velocity is decreasing in a
positive direction, or increasing in a negative
direction.
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- At slope 1 the
object is at rest
- At slope 2 the
object starts from rest and
increases speed at a
constant velocity
- At slope 3 the
object is traveling in a
positive direction at
constant speed
- At slope 4 the
object is slowing down to a
stop at a constant rate while
still travelling north.
y The relationship between acceleration, change in velocity, and time interval
is given by the following equation:
- a= v
t
Ex. A soccer ball travelling at 6.5m/s towards a fence bounces off at 5m/s. If the
ball was in contact with the fence for 0.20s, what is the balls acceleration?
(Towards the fence is positive)
= (-5m/s 6.5m/s)
0.20s
= -4.0m/s = -20m/s2
0.20s
4
3
1
2
Calculating Acceleration
TA
V
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Ex. A truck starting from rest accelerates uniformly to 18m/s [W] in 4.5s. What is
the trucks acceleration?
= 18m/s
4.5s
= 4m/s2 [W]
= vf-vi
(Refer to the diagram) = tf-ti
Ex. A car accelerates from rest at 3.0m/s2 forward for 5.0s. What is the velocity of
the car at the end of 5.0s?
= (3.0m/s2) (5.0s)
= 15m/s *The cars change in velocity after 5.0s is 15m/s forward so in order to
figure out the velocity we must figure out the change in velocity (initial velocity-
final velocity)
15m/s= vf-0: since the cars initial velocity was zero, the answer remains the same
vf =15m/s
Ex. A train is travelling east at 14m/s. How long would it take to increase its
velocity to 22m/s [E] if it accelerated at 0.05m/s2 [E]?
t = 8.0m/s (calculated by getting the change in velocity)
0.50m/s2
= 16s: It would take the train 16s to increase its speed.
Calculating Changes in Velocity and Time
TA
V
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y Objects near the surface of Earth fall to Earth due to the force of gravity.
- Acceleration due to gravity is 9.8m/s2 downward.
- To analyze situations where objects are accelerating due to gravity
(falling) use the vat triangle shown on the previous page.
Ex. Suppose a rock falls from the top a cliff. What is the change in velocity of the
rock after it has fallen for 1.5s? (Down= (-))
v= (a) ( t)
= (-9.8m/s2) (1.5s)
= -15m/s
*Since down is negative, the change in velocity of the rock is -15m/s
y Air resistance is a friction-like force that opposes the motion of objects that
move through the air.
- If an object is falling, air resistance acts upward on the object
- If air resistance were not present, all objects would fall with the same
acceleration (9.8m/s2) regardless of their shape, size, and mass.
Gravity and Acceleration and Calculating Motion due to Gravity