Kinematics

Position, Velocity , and Acceleration Graphs

Overview

Kinematics: A Description of MotionPosition and DisplacementVelocity

AverageInstantaneous

AccelerationAverageInstantaneous

Graphing

The topics we will discover on this powerpoint

A Story

We will introduce Kinematics by looking at motion along a single line. This restricted perspective is usually called 1-Dimensional Motion.

If we can answer the following Questions by using a both Algebra and Graphing, then we know we GROK 1-dimensioal kinematics:

How far did you walk? What where your speeds at any location on your

journey? What where your velocities at any location? What was your average Speed and Average Velocity? What was your Displacement? Tell me about your accelerations during the journey

Imagine walking away from your house to visit an ice cream store that is located 2 km away. You walk at a constant speed and you arrive at the store in 30 minutes. You spend 10 minutes at the store eating your ice cream. You then leave the store and continue walking away from home at your normal pace for 5 minutes. You realise that you should be heading home and immediately walk back towards home at twice your normal pace. At a distance halfway between the ice cream store and your home you stop for 10 minutes at a library (good students always visit libraries). You then continued back home at your increased speed.

Position vs Time PlotsGives us the location, x, at any

time, t.

x

t3

3

-3

1For example:

Position at t=3, x(3) = 1

Graph the Journey

Get out some graph paper and Sketch a Position-Time graph of our story. Don’t forget to use those graphing skills you learned in Math class.

Imagine walking away from your house to visit an ice cream store that is located 2 km away. You walk at a constant speed and you arrive at the store in 30 minutes. You spend 10 minutes at the store eating your ice cream. You then leave the store and continue walking away from home at your normal pace for 5 minutes. You realise that you should be heading home and immediately walk back towards home at twice your normal pace. At a distance halfway between the ice cream store and your home you stop for 5 minutes at a library (good students always visit libraries). You then continued back home at your increased speed.

Imagine walking away from your house to visit an ice cream store that is located 2 km away. You walk at a constant speed and you arrive at the store in 30 minutes

You spend 10 minutes at the store eating your ice cream.You then leave the store and continue walking away from home at your normal pace for 5 minutesYou realise that you should be heading home and immediately walk back towards home at twice your normal pace. At a distance halfway between the ice cream store and your home you stop

stop for 5 minutes at a library (good students always visit libraries).You then continued back home at your increased speed.Compare your graph to mine. Let’s do some algebra to calculate distances and speeds at each line segment of the graph.

• Position (x) – where you are located• Distance (d) – how far you have traveled,

regardless of direction • Displacement (x = xf-xi) – where you are in relation

to where you started, need initial and final position.

Kinematic Definitions

• Displacement is a vector: It has both, magnitude and direction!!

• For one dimensional motion a +ve sign means the displacement is toward the right, a -ve sign means the displacement is toward the left.

• Distance is a scalar, only magnitude.

• Speed (v) – distance divided by time. It is always a positive quantity and the direction of motion is irrelevant. Speed is a scalar quantity.

• Velocity (v) – is displacement divided by time. Since displacement depends only on your starting and ending points, velocity (v) is x divided by t and thus a vector. If displacement is negative, then the velocity will also be negative. Similarly if displacement is positive, then the velocity will be positive.

Kinematic Definitions

,f i

avg xf i

x xxv

t t t

total distance

average speed total time

Graphical Interpretation of Average Velocity Velocity can be determined from a position-time

graph

Average velocity equals the slope of the line joining the initial and final positions (A and D)

40

3.0

13.3

average

x m

t sm

v

s

The average velocity you travelled at on your walk between home and the ice cream store is:

,

2 0 2 10.06667

30min 0min 30min 15min minf i

avg xf i

x xx km km km km kmv

t t t

2km

30 min

We also notice that the total distance travelled by you is also 2km and the total amount of time it took you was also 30 min. So your average speed is also 0.0667 km/min

The average velocity you travelled when sitting at the ice cream store eating your ice cream:

,

2 2 00

30min 40min 10min minf i

avg xf i

x xx km km km kmv

t t t

10 min

We also notice that the total distance travelled by you is also 0 km and the total amount of time it took you was also 10 min. So your average speed is also 0 km/min

The average velocity you travelled for 5 minutes away from the ice cream store:

,

1 12 2 13 3 0.0677

45min 40 min 5min 15 min minf i

avg xf i

km km kmx xx km kmv

t t t

5 min

We also notice that the total distance travelled by you is also 1/3 km and the total amount of time it took you was also 5 min. So your average speed is also 0.0677 km/min

1/3 km

The average velocity you travelled when heading back home:

,

1 11 2 1 23 3 0.1333

55min 45min 10min 15 min minf i

avg xf i

km km kmx xx km kmv

t t t

10 min

We also notice that the total distance travelled by you is also 1 1/3 km and the total amount of time it took you was also 10 min. So your average speed is also 0.1333 km/min

1 1/3 km

Note: Since we are returning home, look what happens to the average speed and average velocity from home to this location:,

1 0 10.01818

55min 0min 55min minf i

avg xf i

x xx km km km kmv

t t t

55 min

We notice that the total distance travelled by you is 2 km +1/3 km + 1 1/3 km = 3.667 km and the total amount of time it took you was also 55 min. So your average speed is now 0.0667 km/min1 km

The average velocity during your library:

,

1 1 0 00

60min 55min 5min 15 min minf i

avg xf i

x xx km km km km kmv

t t t

5 min

We also notice that the distance travelled by you is also 0 km and the total amount of time it took you was also 5 min. So your average speed is also 0 km/min

The average velocity final walk home:

,

0 1 1 20.1333

1 1 15 min min67 min 60min 7 min2 2

f iavg x

f i

x xx km km km km kmv

t t t

7.5 min

We also notice that the distance travelled by you is also 1 km and the total amount of time it took you was also 7.5 min. So your average speed is also 0.1333 km/min

1 km

1

7.5min

0.1333min

ave

kmv

km

The average velocity for the whole journey:

,

0 0 00

1 1 min67 min 0min 67 min2 2

f iavg x

f i

x xx km km km kmv

t t t

67.5 min

We also notice that the distance travelled by you is now 4.667 km and the total amount of time it took you was also 67.5 min. So your average speed is 0.06913 km/min

4.667

67.5min

0.06913min

ave

kmv

km

Understanding

An object (say, car) goes from one point in space to another. After it arrives to its destination, itsdisplacement is:

1. either greater than or equal to 2. always greater than3. always equal to4. either smaller or equal to5. either smaller or larger

than the distance it traveled.

Example:

11

80 10 70

10 107

average

x m m mv

t s sm s

Suppose that in both cases truck covers the distance in 10 seconds:

22

20 80 60

10 106

average

x km km mv

t s sm s

Instantaneous Velocity

This notation tells us that Instantaneous Velocity is a Derivative of position with respect to time. I invented the mathematics for this, and it is called Calculus. Why? Because I need two points to determine a slope, but I have only one point available:

Instantaneous velocity is defined as the limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero

The instantaneous velocity indicates what is happening at every point of time

0 0lim lim f i

instt t

x xx dxv

t t dt

Graphical Interpretation of Instantaneous Velocity

Instantaneous velocity is the slope of the tangent to the curve at the time of interest

The instantaneous speed is the magnitude of the instantaneous velocity

The slope of the tangent line drawn at B, is the Instantaneous Velocity at B

Average vs Instantaneous Velocity

Average velocity Instantaneous velocity

Understanding

The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true:

1. at time tB both trains have the same velocity

2. both trains speed up all the time3. both trains have the same velocity at some time before tB

4. train A is longer than train B5. all of the above statements are true

Note: the slope of curve B is parallel to line A at some point t< tB

A

B

time

position

tB

Average Acceleration

A velocity that changes indicates that an acceleration is present

Average acceleration is the rate of change of the velocity

Average acceleration is a vector quantity (i.e. described by both magnitude and direction)

f iaverage

v vva

t t

Note: Velocity also changes when the speed remains the same, but the direction changes. This is because velocity is a vector.

Average Acceleration

Units

SI Meters per second squared (m/s2)

CGS Centimeters per second squared (cm/s2)

US Customary Feet per second squared (ft/s2)

When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing

When the sign of the velocity and the acceleration are the opposite (either positive or negative), then the speed is decreasing

Instantaneous and Uniform Acceleration

Instantaneous acceleration is the limit of the average acceleration as the time interval goes to zero

When the instantaneous accelerations are always the same, the acceleration will be uniform. That is, the instantaneous accelerations will all be equal to the average acceleration

0 0lim lim f i

instt t

v vv dva

t t dt

Yep, that

Calculus thing again.

Graphical Interpretation of Acceleration

Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph. That is, the slope of secant line PQ

Instantaneous acceleration is the slope of the tangent to the curve of the velocity - time graph, That is, slope of tangent line at t=b

b

Velocity vs Time Plots Gives velocity at any time by just looking at the

height of the graph at any time t. Net area gives displacement (how far you have

moved from initial position to final position) Total area gives how far you have travelled Slope gives acceleration (rise over run). Speed s = | v |

v (m/s)

t4

3

-3

9-1=89+1=10

If the shapes you are calculating the area of is not simple, you can calculate the area using my Calculus.

Let’s Draw a Velocity Time Graph based on out previous ice cream trip’s Position Time graph. Then we will use that to determine how far we walked.

Slope (Velocity) is

0.06667 km/min

Slope (Velocity) is 0

km/min

Slope (Velocity) is 0.06667

km/min Slope (Velocity) is -0.1333 km/min

Slope (Velocity) is 0

km/min

Slope (Velocity) is -0.1333 km/min

Let’s Draw a Velocity Time Graph based on out previous ice cream trip’s Position Time graph. Then we will use that to determine how far we walked.

2 km1/3km

1 1/3km

1 km

Hey, you walked 2 km + 1/3 km= 2 1/3 km away from home and 1 1/3 km+ 1 km = 2 1/3 km back toward home. So our net distance travelled is 0km. While the total distance travelled was 2 1/3 km+ 2 1/3 km = 4 2/3 km

2

6

Acceleration vs Time Plots

Gives acceleration at any time.Area gives change in velocity

a (m/s2)

t4

3

-3

The change in velocity between t=4 and t=1 is then 6+(-2)=4 m/s.

Is it possible for an object to have a positive velocity at the same time as it has a negative acceleration?

1 - Yes

2 - No

Let’s Check your Understanding

If the velocity of some object is not zero, can its acceleration ever be zero ?

1 - Yes 2 - No

Velocity Understanding

If the average velocity of a car during a trip along a straight road is positive, is it possible for the instantaneous velocity at some time during the trip to be negative?

A - Yes

B - No

Drive north 5 miles, put car in reverse and drive south 2 miles. Average velocity is positive while the instantaneous velocity is negative when the car is going backwards..

Graphical Relationships

x

dxv

dt

2

2x

x

dv d xa

dt dt

( )x t

• Relating Graphs

x

v

a

t

t

t

Understanding

Draw vy vs t

A ball is dropped from a height of two meters above the ground.

-6

v

t0.5

9

-6

v

t0.5

9

-6

v

t0.5

9

-6v

t0.5

9v

t0.5

9

-6

A B C

D E

y

x

Understanding

Draw v vs t

Draw x vs t

Draw a vs t

v

t

x

t

a

t

A ball is dropped for a height of two meters above the ground.

Tossed Ball

Draw v vs t

A ball is tossed from the ground up a height of two meters above the ground. And falls back down.

vv v

-6 -6

t1

9

-6

t1

9

-6

t1

9

-6v

t1

9v

t1

9

A B C

D E

y

x

Tossed Ball

Draw v vs t

Draw x vs t

Draw a vs t

v

t

x

t

a

t

A ball is tossed from the ground up a height of two meters above the ground. And falls back down.

= 0

A ball is thrown straight up in the air and returns to its initial position. During the time the ball is in the air, which of the following statements is true?

A - Both average acceleration and average velocity are zero.

B - Average acceleration is zero but average velocity is not zero.

C - Average velocity is zero but average acceleration is not zero.

D - Neither average acceleration nor average velocity are zero.

Understanding

Vave = DY/Dt = (Yf – Yi) / (tf – ti)

aave = DV/Dt = (Vf – Vi) / (tf – ti)aave is not 0 since Vf and Vi are not the same !..

-100

-80

-60

-40

-20

0

20

0 5 10 15 20

v (m/s)

t (seconds)

-300

-200

-100

0

100

0 5 10 15 20

x (meters)

t (seconds)

• Where is velocity zero?• Where is velocity positive?• Where is velocity negative?• Where is speed largest?

• Where is acceleration zero?• Where is acceleration positive?

position vs. time

velocity vs. time

Example

Final Check

Slope of x vs t gives v Area under v vs t gives x!

Which plot best represents v(t)

x(t)

Dt

Dx

v(t)v(t) v(t)

t

t t t

5s

4s

Matching Graphs