Inertial Reference Frames
Concept
Complicated multi-dimensional motions can be analysed
as a superposition of motions along orthogonal axes.
• 2-D kinematics example
• Relative velocity and acceleration
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Physics 1D03 - Lecture 5
Concept Question
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Concept
If you throw a ball on level ground at speed v0 and
angle q, how far away will it land?
•
•
•
•
A coordinate system specifies direction vectors
The coordinate system may be moving
Inertial coordinate systems are not accelerating
An inertial coordinate system is called an inertial
reference frame
• Physics is predictable in an inertial reference
frame
v0
θ
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Quick Quiz 8
Concept
Sitting in a high-speed train moving at a constant
velocity, I drop a cube of sugar into my cup of tea.
What is the trajectory of the sugar cube as it falls
from my hand?
Accelerations measured from different co-ordinate
systems are equal, unless the co-ordinate systems
are accelerating.
(a) It falls straight down into the cup.
(b) It follows a curved trajectory ending up slightly
ahead of where I dropped it.
(c) It follows a curved trajectory ending up slightly
behind where I dropped it.
Physics 1D03 - Lecture 5
x‘
x
5
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Inertial or Non-Inertial?
P
path plotted in
xy co-ordinates
r’(t)
r(t)
Physics 1D03 - Lecture 5
• If v0 is constant, the accelerations measured in the
two coordinate systems are the same.
ÆThey are called inertial reference frames
y’
r0(t)
y
• The position a particle P is described by r(t) in (x,y)
• The same particle is described by r’(t) in (x’,y’)
• r0(t) connects the origins of the 2 coordinate systems.
• If v0 is not constant, the measured accelerations
are different.
ÆThey are called non-inertial reference frames
r(t) = r0(t) + r’(t)
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shoot
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#2 1D03Ball
Ball
shoot
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Quick Quiz 9
How do you test whether or not you are in an inertial
reference frame?
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Physics 1D03 - Lecture 5
Relative velocity and acceleration
How are these velocities related?
Example:The wind is blowing from north to south at
Try an easier problem in one dimension :
Wind blows southwards, bicycle travels north.
30 km/h. You are riding your bicycle northeast at 20
km/h. How fast does the wind feel to you, and which
direction does it seem to blow?
There are two reference frames (coordinate axes)
for measuring from: the ground, and the bicycle.
r
Write: v w ,g for wind relative to ground
r
v b,g for bicycle relative to ground
r
v w ,b for wind relative to bicycle
bicycle
A cyclist riding at 20 km/h directly into
a 30 km/h wind should feel a 50 km/h
headwind.
r
v b,g
But we should write this in terms of
vectors :
10
r
v w ,g
r
v w ,b
r
v b,g
r
r
r
From the diagram: v w ,b = v w ,g − v b,g
wind
r
v w ,g
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The same (vector) equation is valid for all directions.
r
r
r
From the diagram: v w ,b = v w ,g − v b,g
r
A cyclist moving with velocity v b , g
would see the world moving past him
the same as if he was stationary
and
r
the world has velocity - v b , g .
r
N
Back to our original problem:
r
r
r
v w ,b = v w ,g − v b,g
r
v w ,g
r
v w ,b
OR
r
r
r
vw ,b = vw , g + ( −v b , g )
r
v b,g
135°
r
− vb,g
r
So if we want to know vw ,b , we need to subtract v b , g
to give the cyclist zero velocity in his reference frame.
Physics 1D03 - Lecture 5
r
v w ,g
r
v w ,b
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Summary
N
• Inertial reference frames move at constant
velocity relative to each other
• Acceleration is the same in all inertial frames,
r
r
r
velocities obey :v a,b + v b,c = v a,c
θ
30 km/h
r
v w ,b
135°
You can use your favorite method
r
to solve for v w ,b and angle.
20 km/h
Answer : 46.3 km/h at θ = 18.0° West of South
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