particle motion (aka rectilinear motion). vocabulary rectilinear motion –position function...
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Particle Motion(AKA Rectilinear Motion)
Vocabulary• Rectilinear Motion
– Position function– Velocity function
• Instantaneous rate of change (position time)– Speed function
• Absolute value of velocity– Acceleration Function
• Instantaneous rate of change (velocity time)• Speeding up/Slowing down
Particle Motion
• Motion on a line
Moving in a positive direction from the origin
Moving in a negative direction from the origin
Position Function• Horizontal axis:
– time• Vertical Axis:
– position on a line
Moving in a positive direction from the origin
time
position
Moving in a negative direction from the origin
Position function: s(t)s = position (sposition duh!)t = times(t)= position changes as time changes
Example
• Use the position and time graph to describe how the puppy was moving
time
position
Velocity• Rate
– position change vs time change
– Velocity can be positive or negative• positive: going in a
positive direction• negative: going in
a negative direction
18
16
14
12
10
8
6
4
2
-2
-4
-6
-8
-10
p
1 2 3 4 5 6 7 8 9 10 11 12t
position
time
A A
18
16
14
12
10
8
6
4
2
-2
-4
-6
-8
-10
-12
p
-1 1 2 3 4 5 6 7 8 9 10 11t
v(t) x = 3x2+-34x+76
4
time
Animate Points
Velo
city
Pos
ition
Velocity
• Rate at which a coordinate of a particle changes with time
• s(t) = position with respect to time• Instantaneous velocity at time t is:
dtdststv )(')(
time
position
v(t) = positive – increasing slope – moving in a positive direction
v(t) = negative– decreasing slope – moving in a negative direction
Velocity function
• Velocity is the slope of the position function (change in position /change in time)
• velocity =
– This is instantaneous rate of change (position time)
dtdstv )( )(ts
Position Velocity MeaningPositive Slope Positive y’s moving in a positive direction
Negative slope
Negative y’s Moving in a negative direction
Practice• Let s(t)= t3-6t2 be the position function of a
particle moving along an s-axis were s is in meters and t is in seconds. – Graph the position function– On a number line, trace the path that the particle
took. – Where will the velocity be positive? Negative?– Graph the velocity function– Identify on the velocity function when the particle was
heading in a positive direction and when it was heading in a negative direction.
Example: s(t)= t3-6t2 position
time
23 6)( ttts
time
velocity
tttv 123)( 2
tttv 123)( 2
time
speed
Velocity vs Speed
• Speed is change in position with respect to time in any direction
• Velocity is the change in position with respect to time in a particular direction– Thus – Speed cannot be negative – because
going backwards or forwards is just a distance– Thus – Velocity can be negative – because
we care if we go backwards
Speed
• Absolute Value of Velocity
dtdstv
)(
speedousinstantane
example: • if two particles are moving on the same coordinate line • with velocity of v=5 m/s and v=-5 m/s,• then they are going in opposite directions• but both have a speed of |v|=5 m/s
Practice
• Graph the velocity function • What will the speed function look like?• At what time(s) was the particle heading in
a negative direction? Positive direction?
19163)( 2 tttv
Acceleration
• the rate at which the velocity of a particle changes with respect to time.– If s(t) is the position function of a particle
moving on a coordinate line, then the acceleration of the particle at time t is:
dtdvta )(
2
2
)(")(')( dtsdts
dtdstvtaOR
**The second derivative of the position function!!
Example
• Let s(t) = t3 – 6t2 be the position function of a particle moving along an s-axis where s is in meters and t is in seconds. Find the instantaneous acceleration a(t) and shows the graph of acceleration verses time
tttstv 123)(')( 2 126)('')(')( ttstvta
Speeding Up & Slowing Down
• Speeding up velocity and acceleration are the same sign.
• Slowing down when velocity and acceleration are opposite signs.
Example
• When is s(t) speeding up and slowing down?
position
time
23 6)( ttts
time
velocity
tttv 123)( 2
acceleration
Velocity & Acceleration Functions20
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8
6
4
2
-2
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p
-1 1 2 3 4 5 6 7 8 9 10 11t
Animate Points
A AB
Slowing down
Velocity +
Acceleration -
Speeding up
Velocity -
Acceleration -
Slowing down
Velocity -
Acceleration +
Speeding up
Velocity +
Acceleration +
Analyzing MotionGraphically Algebraically Meaning
Position
Velocity
Acceleration
Positive “s” values Positive side of the number line
Negative side of the number line
Negative “s” values
s(t)=velocity.
Look for Critical PtsPostive “v” values0 “v” values (CP)
Negative “v” values
Moving in + directionTurning/stopped
Moving in a – direction
v(t)=accelerationLook for Critical Pts
+ a, + v = speeding up- a, - v = speeding up+ a, - v = slowing down- a, + v = slowing down
ExampleSuppose that the position function of a particle moving on a coordinate line is given by s(t) = 2t3-21t2+60t+3 Analyze the motion of the particle for t>0
Graphically Algebraically Meaning
Pos
ition
Velo
city
Acc
eler
atio
n
0360212)( 23 tttts
Never 0 (t>0), always postive
Always on postive side of number line
060426)()( 2 tttvts0)107(6 2 tt
0)5)(2(6 tt
0 2 5
+ - +0 0
0<t<2 going pos directiont=2 turning2<t<5 going neg. directiont=5 turningt>5 going pos. direction
t=0 t=2t=5
04212)()( ttatv4212 t 5.3t
+ - - +0 2 53.5
va - - + +
0<t<2 slowing down2<t<3.5 speeding up3.5<t<5 slowing down
5<t speeding up
Applications: Gravity•
• s = position (height)• s0= initial height• v0= initial velocity• t = time• g= acceleration due to gravity
– g=9.8 m/s2 (meters and seconds)– g=32 ft/s2 (feet and seconds)
200 2
1 gttvss
s0
Example• Nolan Ryan was capable of throwing a baseball at 150ft/s (more
than 102 miles/hour). Could Nolan Ryan have hit the 208 ft ceiling of the Houston Astrodome if he were capable of giving the baseball an upward velocity of 100 ft/s from a height of 7 ft?
2161007 tts tv 32100 the maximum height occurs when velocity = 0
t=100/32=25/8 seconds
s(25/8)=163.25 feet
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