Transcript Lecture 3 - McMaster Physics and Astronomy

Kinematics in One Dimension
• Displacement, velocity, acceleration, free fall
• Examples
Knight: Chapters 1, 2
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1-D motion can be described by scalars (real numbers
with units) as functions of time:
Position
x(t)
(displacement from the origin)
Velocity
v(t)=dx/dt (rate of change of position)
Acceleration a(t)=dv/dt (rate of change of velocity)
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A Special Case: Constant Acceleration
dv
dx
, v
Using the definitions a 
we can derive
dt
dt
a  constant
v(t )  vo  at
x(t )  vo t 
1
2
Caution: These assume
acceleration is constant.
at
2
From the above you can get:
v2  v1  2ad
2
2
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Example: Free Fall.
(“Free fall” means the only force is gravity; the motion can
be in any direction).
All objects in free fall move with constant downward
acceleration:
a  g  9.80 m / s 2 [downwards]
This was demonstrated by Galileo around 1600 A.D.
The constant “g” is called the “acceleration due to gravity”.
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Quix 1
A block is dropped from rest. It takes a time t1 to
fall the first third of the distance. How long does it
take to fall the entire distance?
a) 3 t1
b) 3t1
c) 9t1
d) None of the above
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Example 1
A particle’s position is given by the
function:
x(t)=(-t3+4t) m
a) what is the velocity at t=3 s ?
b) what is the acceleration at 3 s ?
c) make a sketch of the motion
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Example 2
An object if thrown straight up with a
velocity of 5m/s. What will the velocity
be when it comes back to its original
position ?
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Example 3
A skier is moving at 40m/s at the top of a
hill. His velocity changes to 10m/s after
covering a distance of 600m. What is
his acceleration ?
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Example 3b
The skier’s girlfriend is also traveling at
40m/s, but, unfortunately, after only
3s, hits a tree and her velocity
‘suddenly’ comes to 0m/s.
How far did she get, given the same
deceleration as in the previous
question?
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Vector Review
• Scalars and Vectors
• Vector Components and Arithmetic
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Physical quantities are classified as scalars,
vectors, etc.
Scalar : described by a real number with units
examples: mass, charge, energy . . .
Vector : described by a scalar (its magnitude)
and a direction in space
examples: displacement, velocity, force . . .
Vectors have direction, and obey
different rules of arithmetic.
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Notation
• Scalars : ordinary or italic font (m, q, t . . .)
• Vectors : - Boldface font (v, a, F . . .)
  
- arrow notation ( v, a, F . . .)
- underline (v, a, F . . .)
• Pay attention to notation :
“constant v” and “constant v” mean
different things!
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Coordinate Systems
In 2-D : describe a location in a plane
y
• by polar coordinates :
(x,y)
distance r and angle 
r
y
• by Cartesian coordinates :

0
x
x
distances x, y, parallel to axes
with:
x=rcosθ y=rsinθ
These are the x and y components of r
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Addition:
If A + B = C ,

A
Ay
Ax

B
By
Bx
then:
Cx  Ax  Bx
Cy  Ay  By
Cz  Az  Bz
Tail to Head

B

C
Three scalar
equations from one
vector equation!
Cy
Bx

A
Ax
By
Ay
Cx
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