Kinematics in One Dimension
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Displacement, velocity, acceleration
Graphs
A special case: constant acceleration
Bodies in free fall
Serway and Jewett Chapter 2
Physics 1D03 - Lecture 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) (rate of change of position)
Acceleration a(t) (rate of change of velocity)
•The sign (positive or negative) keeps track of direction (in 1-D).
• Algebraic relations involving position, velocity, and acceleration
come from calculus.
• The same relations can be seen from graphs of position, velocity,
and acceleration as functions of time.
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• Kinematics : the description of motion
• One dimension : motion along a straight
line (e.g., the x-axis)
Examples - sprinter running 100 meters in a straight line
- ball falling straight down, and bouncing back up
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Displacement : x  x 2  x 1
x
position x as a function of time t
x2
x
x1
t
t1
t2
Average velocity : v
t
 x / t
(slope of the line)
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Instantaneous velocity is the average over an
‘infinitesimal’ time interval :
x dx
t 2  t 1 , t  0 and
 v
t
dt
x
t
t
v is the slope of the tangent to the x vs. t graph.
Physically, v is the rate of change of x, hence dx/dt.
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Acceleration is the rate of change of velocity:
Average Accelerati on : a 
v v2  v1

t t 2  t1
dv
Instantane ous Accelerati on : a 
dt
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Graphs of x(t), v(t), a(t)
position x
acceleration a
time
velocity v
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Quiz
A rubber ball is dropped and bounces twice from the floor
before it is caught. (Take x to be upwards, and x=0 at the
floor.)
At the highest point of the first bounce, v and a are:
a) both nonzero
b) one is zero, one is not zero
c) both zero
d) other (explain)
Suggestion: Sketch graphs of x, v, a vs. time.
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Quiz
A particle (in one dimension) is initially moving.
A few seconds later it has stopped (not moving).
During that time interval:
a) The particle’s average acceleration is positive
b) The particle’s average acceleration is negative
c) Not enough information to tell
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A Special Case: Constant Acceleration
dv
dx
, v
Use the definitions a 
and derive
dt
dt
a  constant
Caution: These assume
acceleration is constant.
v(t )  at  v0
x(t ) 
1
2
a
t
 v0t  x0
2
Exercise: eliminate t or a to show that
v 2  v0  2a( x  x0 )
2
v  v0 x  x0

v
2
t
These are sometimes convenient,
but not necessary. They are valid
only for constant acceleration.
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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.
“g” is called the “acceleration due to gravity” or the
“gravitational field of the Earth”.
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The free-fall acceleration is the same for all
objects; size and composition don’t matter.
But:
• g varies slightly with location and height, about
0.03 m/s2 over the surface of the Earth, and up to
a few kilometers above
• if air resistance is significant, we don’t really have
“free fall”.
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Quiz
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) 3t1
b) 3t1
c) 9t1
d) None of the above
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Quiz
You throw a set of keys up to a window 4.9m above
you. If the keys just make it to your friend on a balcony
1.0s later, what was their initial velocity ?
a) 40 m/s
b) 9.8 m/s
c) 4.5 m/s
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