Chapter 2 – Describing Motion
Motion can be deceiving. In order to describe motion in a meaningful way, you have to establish a frame of
reference.
A frame of reference is something that a person uses to compare the position or motion of an object to. The
motion of the object can be completely different depending on the frame of reference used.
ie. A person driving down the road in their car at 100 km/h.
Frame of Reference 1 – the car itself
- in relation to the car, the person is not moving
Frame of reference 2 - the road
- in relation to the road, the person is going 100km/h.
Frame of Reference 3 – the sun
- in relation to the sun, the person is moving at over 100 000 km/h.
Illustrating Motion
If you have a series of diagrams of an object, or a series of dots, showing the position of the object after equal
intervals of time, you can determine the details of its motion.
ie.
Situation 1 – If the distance is the same between each diagram, then the object is moving at a constant speed
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Situation 2 – If the distance between is increasing, then the object is speeding up.
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Situation 3 – If the distance between is decreasing, the object is slowing down.
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Displacement and Velocity
In the previous diagrams, only the quality of the objects motion have been described. (qualitative descriptions)
In physics we are also interested in describing motion quantitatively. (using measurements)
However if you knew the elapsed time between each position of the object, you could determine its position,
speed and rate of change of speed.
Vectors and Scalars
Most measurements are scalar quantities or scalars. Scalars are quantities which only need a magnitude to
completely describe them.
ie. mass, time, distance , speed, length, temp., moles, amps,
In physics, motion is usually described by vector quantities or vectors. Vectors need both a magnitude and a
direction to completely describe them.
ie. weight, force, position, displacement, velocity, acceleration
A vector is represented by an arrow drawn in a frame of reference. The length of the arrow represents the
magnitude (when accompanied by a scale), and the direction the arrow points describes the direction.
tail
tip
In order for a vector to mean anything there must be a scale associated with it.
Position Vectors
position (d) – is the separation between an object and some reference point.
- Position needs a direction and therefore is a vector.
- An object to the right of the reference point is assigned a (+) direction and those to the left are given a (-)
direction.
Position Vector – points from the origin of a coordinate system to the location of an object at a particular
instant in time.
distance – is the separation between two points. No frame of reference or direction is needed, therefore it is a
scalar quantity.
Displacement
displacement (Δd)– the vector difference between the final position and the initial position. (measurement of
an objects change in position.)
d1 – position at 1st clock reading d2 - position at 2nd clock reading
Δd = change in position = d2 – d1
The displacement of the object does not depend on what takes place between the initial and final positions, it
only depends on those two positions.
ie. an object thrown 10 m into the air.
d1 – ball in the person’s hand before throwing
d2 – ball in the person’s hand after it is caught
Δd – 0 m. There is no displacement even thought the ball traveled 20 m.
average velocity – a measure of a bodies displacement over the time it took for the displacement to occur.
v = ∆ d = d2 – d1
∆t
t2 – t1
Average velocity is usually measured in m/s.
If the object has the same average velocity over all measures time intervals, it is said to have a constant
velocity.
Position – Time Graphs
This is a visual representation that shows how the position of a body is related to the clock reading.
At this point in time we will only be dealing with bodies that have a constant or uniform velocity.
slope = rise = ∆ Y = ∆ d = m = v
run
∆X
∆t
s
The steeper the slope, the greater the objects velocity.
Since positions can be +ve or –ve, therefore displacements can also be +ve or –ve. From this we see that
velocities can also be –ve.
A –ve velocity indicates that the object is moving in the opposite direction of that of an object with a +ve
velocity.
Interpreting a Graph
+ve slope = +ve velocity
-ve slope = -ve velocity
0 slope = 0 velocity
Instantaneous Velocity
The velocity of an object usually varies over any given time interval. ie. sprinter
instantaneous velocity – this is the velocity of an object at a particular instant in time. (clock reading)
We have seen that the slope of a position – time graph gives us the object’s velocity.
For a non-linear graph, to find the instantaneous velocity, we must find the slope of the tangent to the graph at
that particular moment in question.
Velocity – Time Graphs (Constant Velocity)
Since the velocity is constant, all values of time will have the same plotted value for the velocity.
This produces a horizontal straight line graph.
velocity (km/h)
250
200
150
100
50
0
0
1
2
3
time (h)
4
5
The area under the graph allows us to calculate the displacement of the object for any time interval.
ie. refer to the previous graph.
Displacement of objects after 2 hrs.
25 km/h
100 km/h
200 km/h
a=lxw
a = 2h x 25 km/h
a = 50 km
a=lxw
a = 2 h x 100 km/h
a = 200 km
a=lxw
a = 2 h x 200 km/h
a = 400 km
since l = time and w = velocity , therefore a = displacement
Average acceleration – the change in an object’s velocity over the time interval which it occurred.
Change in velocity = ∆v = v2 – v1
Time interval = ∆t = t2 – t1
... a = ∆v = v2 – v1
∆t
t2 – t1
Acceleration is measured in m/s = m/s2
s
ie. a = 10 m/s2 - the object’s velocity increases by 10 m/s for
every second of motion.
Time (s)
0.0
1.0
2.0
3.0
4.0
Velocity (m/s)
0.0
10.0
20.0
30.0
40.0
Since most velocity is a vector, therefore acceleration is also a vector.
In most cases, an object with a +ve acceleration is increasing its velocity over time and one with a –ve
acceleration is decreasing its velocity over time.
* exception – if an object is going in a –ve direction, and is slowing down, it will have a +ve acceleration
and if it is speeding up going in a –ve direction, it will have a –ve acceleration.
ie. a = ∆v = v2 – v1
∆t
t2 – t1
= (-5) – (-10)
2.0
v1 = -10 m/s
=
v2 = -5 m/s
+5
2
∆t = 2 s
= +2.5 m/s2
Velocity – Time Graph of a Uniformly Accelerated Object
For uniform acceleration, the graph will be a straight line, therefore showing that the change in velocity is the
same for each interval of time.
This means that the speed is proportional to the time traveled.
ie.
velocity (m/s)
80
60
15m/s2
40
10 m/s2
5 m/s2
20
0
0
1
2
3
4
time (s)
We can see that the slope of the line gives us the acceleration.
The greater the slope of the graph, the greater the rate of acceleration.
+ve slope
+ve acceleration
- ve slope
- ve acceleration
0 slope
0 acceleration
constant velocity
If the acceleration is constantly changing than so will the slope of the graph, resulting in a graph that is not a
straight line.
Average vs. Instantaneous Acceleration
average acceleration – is the change in velocity measured over a definite time interval.
instantaneous acceleration – is the acceleration of an object at a particular moment in time.
It is the slope of the tangent to the velocity-time graph.