3.4 b) Particle Motion / Rectilinear Motion
Our Lady of Lourdes
Math Department
Rectilinear 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 (s position)t = times(t)= position changes as time changes
Example 1
• The figure below shows the position vs. time curve for a particle moving along an s-axis. In words, describe how the position of the particle is changing with time.
Example 1
• The figure below shows the position vs. time curve for a particle moving along an s-axis. In words, describe how the position of the particle is changing with time.
• At t = 0, s(t) = -3. It moves in a
positive direction until t = 4
and s(t) = 3. Then, it turns around
and travels in the negative
direction until t = 7 and s(t) = -1.
The particle is stopped after that.
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 12
t
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 11
t
v(t) x = 3x2+-34x+76
4
time
Animate Points
Vel
ocity
Pos
ition
Velocity function
• Velocity is the slope of the position function (change in position /change in time)
• velocity =
– Technically, this is instantaneous velocity
dt
dstv )( )(ts
Position Velocity Meaning
Positive Slope Positive y’s moving in a positive direction
Negative slope Negative y’s
Moving in a negative direction
Velocity or Speed
• Speed: 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
dt
dstv
)(
speed
ousinstantane
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
Example 2
• Let s(t) = t3 – 6t2 be the position function of a particle moving along the s-axis, where s is in meters and t is in seconds. Find the velocity and speed functions, and show the graphs of position, velocity, and speed versus time.
Example 2
• Let s(t) = t3 – 6t2 be the position function of a particle moving along the s-axis, where s is in meters and t is in seconds. Find the velocity and speed functions, and show the graphs of position, velocity, and speed versus time.
ttdt
dstv 123)( 2 |123||)(| 2 tttvspeed
Example 2
• The graphs below provide a wealth of visual information about the motion of the particle. For example, the position vs. time curve tells us that the particle is on the negative side of the origin for 0 < t < 6, is on the positive side of the origin for t > 6 and is at the origin at times t = 0 and t = 6.
Example 2
• The velocity vs. time curve tells us that the particle is moving in the negative direction if 0 < t < 4, and is moving in the positive direction if t > 4 and is stopped at times t = 0 and t = 4 (the velocity is zero at these times). The speed vs. time curve tells us that the speed of the particle is increasing for 0 < t < 2, decreasing for 2 < t < 4 and increasing for t > 4.
Acceleration
• The rate at which the instantaneous velocity of a particle changes with time is called instantaneous acceleration. We define this as:
dt
dvtstvta )()()( ///
Acceleration
• The rate at which the instantaneous velocity of a particle changes with time is called instantaneous acceleration. We define this as:
• We now know that the first derivative of position is velocity and the second derivative of position is acceleration.
dt
dvtstvta )()()( ///
Example 3
• 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 acceleration function a(t) and show that graph of acceleration vs. time.
Example 3
• 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 acceleration function a(t) and show that graph of acceleration vs. time.
tttvts 123)()( 2/ )2(6
126)()(//
t
ttats
Speeding Up and Slowing Down
• We will say that a particle in rectilinear motion is speeding up when its speed is increasing and slowing down when its speed is decreasing. In everyday language an object that is speeding up is said to be “accelerating” and an object that is slowing down is said to be “decelerating.”
• Whether a particle is speeding up or slowing down is determined by both the velocity and acceleration.
The Sign of Acceleration
• A particle in rectilinear motion is speeding up when its velocity and acceleration have the same sign and slowing down when they have opposite signs.
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 Pts
Postive “v” values
0 “v” values (CP)
Negative “v” values
Moving in + direction
Turning/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
Example
Suppose 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
Vel
ocity
Acc
eler
atio
n
0360212)( 23 tttts
Never 0 (t>0), always positive
Always on postive side of number line
060426)()( 2 tttvts0)107(6 2 tt0)5)(2(6 tt
0 2 5
+ - +0 0
0<t<2 going pos direction
t=2 turning
2<t<5 going neg. directiont=5 turning
t>5 going pos. direction
t=0 t=2t=5
04212)()( ttatv4212 t 5.3t
+ - - +
0 2 53.5
va - - + +
0<t<2 slowing down
2<t<3.5 speeding up
3.5<t<5 slowing down
5<t speeding up