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