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On a line. 3024 Rectilinear Motion. AP Calculus. Position. Defn : Rectilinear Motion : Movement of object in either direction along a coordinate line (x-axis , or y-axis) s(t) = position function - position versus time graph (historical note: - PowerPoint PPT Presentation
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3024 Rectilinear Motion
AP Calculus
On a line
Position
Defn: Rectilinear Motion: Movement of object in either direction along a
coordinate line (x-axis , or y-axis)
s(t) = position function - position versus time graph (historical note:
x(t) = horizontal axis y(t) = vertical axis
a directed distance (a vector quantity) of the particle from some point,
p, at instant t . negative time = time before
s(t) positive – the particle is_______________________________________
negative – the particle is _______________________________________
= 0 – the particle is _______________________________________ _
physi
cs
math
SpotiaLatin
Direction and quantity
Located to the right
Located to the left
At the origin
Velocity
v(t) = velocity function - the rate of change of position
Velocity gives both quantity of change and direction of change (again a vector quantity) Speed finds quantity only. - absolute value of velocity (a scalar quantity)
Rem: Average Velocity = change in position over change in time =
= Instantaneous Velocity the derivative
st
0t
sLimt
( ) ( )v t s t
Magnitude only
𝑥 (𝑏) −𝑥(𝑎)𝑏−𝑎
Average speed
𝑥 ′ (𝑡 ) Ticketed speed
Velocity
v(t) = velocity function - the rate of change of position
= Instantaneous Velocity the derivative
v(t) positive – the particle’s position is ____________________ < velocity in a positive direction - _______________________
negative – the particle’s position is _____________________< velocity in a negative direction - _______________________
= 0 - the particle is _____________________________
{This is the 1st Derivative Test for increasing /decreasing!}
0t
sLimt
( ) ( )v t s t
First derivative is positive y is increasing
increasing
Moving to the rightdecreasing
Moving to the leftstationary not moving
*
Acceleration
a(t) = acceleration function - rate of change of velocity
a(t) positive - velocity is __________________________________ < acc. in a positive direction – _________________________
negative - velocity is __________________________________ < acc. in neg. direction – ______________________________
= 0 - velocity is __________________________________
{This is the 2nd derivative test for concavity}
CAREFUL: This is not SPEEDING UP or SLOWING DOWN!
( ) ( )( ) ( ) ( )
v t a ts t v t a t
increasing
Pushed to the rightdecreasing
Pushed to the left
constant cruise control
Speed and Direction
Determining changes in Speed
speed increasing if v(t) and a(t) have same sign -
also for v(t) = 0 and a(t) 0
speed decreasing if v(t) and a(t) have opposite signs -
Determining changes in Direction
direction changes if v(t) = 0 and a(t) 0
no change if both v(t) = 0 and a(t) = 0
Moving Pushed
Moving Pushed OrMoving Pushed
Ball bouncing
Sitting still
Method (General):
1) Find the Critical Numbers in First and Second Derivatives.
1) Answer any questions at specific locations.
2) Do the Number Line Analysis (Brick Wall).
1) Find direction - moving , pushed , and speed
3) Identify the Change of Direction locations
1) Find values at beginning, ending, and change of direction times.
4) Sketch the Schematic graph.
5) Find the Displacement and Total Distance Traveled.
Set = 0
Set = 0
Example: A particle’s position on the y –axis is given by:
4 2( ) 6 6 [ 3,3]y t t t t
1) Find y(t), v(t) and a(t) at t = 2. Interpret each value.
𝑚𝑒𝑡𝑒𝑟𝑠
𝑡𝑖𝑚𝑒𝑖𝑛𝑠𝑒
𝑐𝑜𝑛𝑑𝑠
𝑦 (𝑡 )=𝑡 4 −6 𝑡 2− 6 𝑦 (2 )=16 − 24 −6=− 14
𝑣 (𝑡 )=𝑦 ′ (𝑡 )=4 𝑡 3 −12𝑡 𝑦 ′ (2 )=32− 24=8
𝑎 (𝑡 )=𝑦 ′ ′ (𝑡 )=12 𝑡2 −12 𝑦 ′ ′ (2 )=48−12=36
Located 14 units left
Moving right 8 units/sec
Pushed 36 units2/sec
Speed is increasing
Example: A particle’s position on the y –axis is given by:4 2
3 2
( ) 6 6 [ 3,3]
( ) 4 12 ( ) 12 12
y t t t t
v t t t a t t
2) Determine the motion within each interval: location, direction moving
and direction pushed.
3) Find the values at t = -3, t = 3 and where the particle changes directions.
4) Find the Displacement and Total Distance Traveled.
v(t)
a(t)
m
P
s
-3 −√3 √3 30
neg pospos neg
-2 -1 1 2
Moving left
4 𝑡 (𝑡2− 3) 𝑡=0 𝑡=±√3
𝑦 (− 3 )=21𝑦 (−√3 )=−15
𝑦 (3 )=21
𝑦 (0 )=6𝑦 (√3 )=−15
>36> 9>36>9
Original equation y values
36+9+9+36Ending – beginning 21-21=0
Number line analysis
21
6
-15
Example 2 : A particle’s position on the x –axis is given by:3 2( ) 12 36 24 [ 1,9]x t t t t t
Find and interpret x(t), v(t), and a(t) at t = 5
Example: A particle’s position on the x –axis is given by:
Sketch:
v
a
mps
3 2( ) 12 36 24 [ 1,9]x t t t t t
v
a
mps
𝑠 (𝑡 )=3 𝑡2 −12 𝑡+1 ; [0,5 ]
v
a
mps
𝑠 (𝑡 )=𝑡2+3 𝑡− 6 ; [− 2,2]
v
a
mps
𝑠 (𝑡 )=𝑡3 −9 𝑡+1 ;[−3,3 ]
v
a
mps
𝑠 (𝑡 )=−2 𝑡3+15 𝑡 2− 24 𝑡− 6 ;[0,5 ]
v
a
mps
𝑠 (𝑡 )=24=6 𝑡− 𝑡3
v
a
mps
v
a
mps
Last Update
• 11/22/10
• Assignment: work sheet - Swokowski