Linear Motion
Concept of motion
We shall restrict this discussion to the movement of
an object in a straight line (i.e. 1 D).
The position of an object on a straight line can be
defined by a single number telling us:
(a) how far the object is from a given point
called the origin.
(b) whether the object is on the RHS (Positive
+ve ) or LHS (negative –ve) of the origin.
• When an object moves it undergoes a
Displacement, this is the change in position.
Rate is a quantity divided by time.
Motion is measured relative to an object
assumed stationary.
Conceptual Physics - 3rd edition – Paul Hewitt
Chapter 2 – Linear Motion
Page 1 of 11
Speed is:
• a measure of how fast an
object is moving.
• the rate at which distance
is covered.
• the distance travelled per unit time
speed =
distance d
=
time
t
the units are
(metre)
(second)
Units are in metre/second or metres per sec,
i.e. m/s or m·s-1
Instantaneous Speed
Is the speed of the object at an instant in time.
e.g. the reading on a car’s speedometer.
Speed = 20 km/h or 20 km h-1
Conceptual Physics - 3rd edition – Paul Hewitt
Chapter 2 – Linear Motion
Page 2 of 11
Speed & Velocity
Average Speed
average speed =
total distance moved
time taken
The units are m·s-1
Velocity
Speed and velocity are often interchanged but in
physics velocity is speed in a given direction.
Note: the speed of a car is measured by a
speed-o-meter not a velocity-o-meter!
speed 70 km/h. velocity 70 km/h East.
eg.
Constant Speed
Motion with the same (i.e. a constant) speed.
Constant Velocity
This is motion in a given direction in a straight line
at constant speed.
Discussion – a car is moving in a circle?
The car moves with
constant speed.
The car has changing
velocity.
Explain why this is so?
Conceptual Physics - 3rd edition – Paul Hewitt
Chapter 2 – Linear Motion
Page 3 of 11
Acceleration
Acceleration
- the rate at which velocity is changing
change in velocity v 2 - v 1 ∆v
acceleration =
=
=
time interval
t 2 - t1
∆t
(N.B. The symbol ∆ (delta) is often used as a
symbol for “change in” or “difference in”.
i.e. ∆v means change in velocity).
metres/second (velocity)
Units =
second (time)
metres
metres
=
=
second × second
second 2
= m ⋅ s − 2 or m/s 2
The key idea is that acceleration defines change.
Conceptual Physics - 3rd edition – Paul Hewitt
Chapter 2 – Linear Motion
Page 4 of 11
Acceleration applies also when a change in
direction occurs for example moving around a
circle at constant speed involves acceleration.
(Your velocity is not constant because you
are changing direction.)
Acceleration
increasing
speed
Acceleration
negative
decreasing
speed
Acceleration
changing
direction
Acceleration: since it is defined as the rate of
change of velocity, acceleration has the same
direction as the change in velocity.
Conceptual Physics - 3rd edition – Paul Hewitt
Chapter 2 – Linear Motion
Page 5 of 11
Free Fall
Free Fall - how fast?
An object is dropped.
It starts from rest (v0 = 0 m·s-1) and gains speed
as it falls. It accelerates.
Gravity causes it to gain speed (accelerate).
If there is no air resistance then the object is said
to be in "Free Fall".
Free Fall :- motion under the influence of
gravitational force alone (no air resistance).
Increasing velocity
Constant acceleration (g)
The acceleration of an object under free fall on
Earth is 9.8 m·s-2 (approx. 10 m·s-2).
g = acceleration due to gravity
g = 10 m·s-2
Conceptual Physics - 3rd edition – Paul Hewitt
Chapter 2 – Linear Motion
Page 6 of 11
change in velocity v - v 0
(1) acceleration = time interval = ∆t
if
v = instantaneous speed after an elapsed time
∆t = time interval = elapsed time
v0 = initial speed = 0 m·s-1
acceleration = g m·s-2
v-0
g=
then
elapsed time
∴ v = g x elapsed time
(instantaneous speed)
v = gt
(2) If an object is thrown vertically upwards,
at the top of its flight its velocity is zero
(acceleration is constant since this is the
rate of change of velocity).
At the top of the path
a = 10 m·s-2 (down),
v = 0 m·s-1.
Conceptual Physics - 3rd edition – Paul Hewitt
Chapter 2 – Linear Motion
Page 7 of 11
Free Fall - how far?
( for a body starting from rest)
From before, v = gt
v0 + v
Average speed =
2
and
Distance = average speed × time interval
v+0
×t
∴ d=
2
(gt)
×t
d=
2
1
d = gt
2
2
Conceptual Physics - 3rd edition – Paul Hewitt
Chapter 2 – Linear Motion
Page 8 of 11
Graphs of Motion
(For an object moving in a straight line)
Motion can be presented graphically.
Motion graphs have velocity plotted on the y-axis
(vertical axis) and time plotted along the x-axis
(horizontal axis)
for example:
-1
Velocity (m.s )
50
40
30
20
10
0
0
1
2
3
4
5
T im e (s)
This graph shows that there is a linear relationship
between velocity and time. Also the graph passes
though the origin and hence we can say that
velocity is directly proportional to time, or v ∝ t.
Conceptual Physics - 3rd edition – Paul Hewitt
Chapter 2 – Linear Motion
Page 9 of 11
The slope (gradient) of a straight line graph is
the y step divided by the x step, or rise over run.
For example the gradient (slope) calculation for:
(5 s , 50 m·s-1) and (1 s, 10 m·s-1).
slope =
y step ∆ v
=
x step ∆ t
50 m ⋅ s −1 − 10 m ⋅ s −1
=
5 s − 1s
40 m ⋅ s −1
=
= 10 m ⋅ s −2
4s
So the gradient tells us that the object has an
acceleration of 10 m·s-2.
The equation of a straight line is:
y = mx + c
where m is the gradient (slope) of the line
and c is the y-intercept.
From the graph, m = 10 and c = 0,
therefore y = 10x.
Or, in this case, v = 10t,
where v is the velocity and t is time.
Conceptual Physics - 3rd edition – Paul Hewitt
Chapter 2 – Linear Motion
Page 10 of 11
A curved line on a graph tells us that the
gradient (whatever the units) is constantly
changing with time.
Displacement (m)
35
30
25
20
15
10
5
0
-5 0
1
2
3
4
5
6
Time (s)
This graph has the equation d = t2, with the
black lines indicating the tangent to the curve.
The units of the slope of this curve would be m·s-1
(y over x), or velocity.
The slope of the tangent to the curve increases
with time (the tangent at t = 5 s has a steeper slope
than that at t = 1 s), therefore velocity increases
with time, i.e. the object is accelerating.
Conceptual Physics - 3rd edition – Paul Hewitt
Chapter 2 – Linear Motion
Page 11 of 11