SPH4C1 Lesson 02 Motion and its Applications

1

NON-UNIFORM MOTION

LEARNING GOALS

Students will:

Determine instantaneous and average velocities from position-time graphs.

Describe in words motions represented by position-time graphs.

Learn how to read values, find slopes and calculate areas on graphs.

Learn what these values mean on both position-time and velocity-time graphs.

INSTANTANEOUS VELOCITY

The moment-to-moment measure of an object’s velocity is called its instantaneous velocity. A

vehicle speedometer tells you the instantaneous velocity.

Uniform motion or constant velocity is motion at a constant speed in a straight line. It is the

simplest type of motion that an object can undergo, except for being at rest. In contrast, motion

with non-uniform velocity is motion that is not at a constant speed or not in a straight line.

Motion with non-uniform velocity may also be called accelerated motion.

DETERMINING INSTANTANEOUS VELOCITY FROM A POSITION-TIME GRAPH

If the velocity changes every moment during the motion of an object, no portion of the position-

time graph is a straight line. Remember that each point on the curve represents the position of

the object at each instant in time. To determine the velocity of an object at any instant,

physicists use tangents. A tangent is a straight line that touches a curve at only one point and

is parallel to the curve at that point. Each tangent has a unique slope that represents the

instantaneous velocity at that instant in time.

A. In each of the position-time graph examples below, state whether the object is speeding

up or slowing down.

B. In each of the examples above (in #3), state whether the object is going forwards or

backwards.

C. In the graph below one line is a Ferrari and the other is a Toyota. Which line is most

likely to be the Ferrari? Explain.

SPH4C1 Lesson 02 Motion and its Applications

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EXAMPLE 1 INSTANTANEOUS VELOCITY ON AN POSITION-TIME GRAPH

The position-time data for an all-terrain vehicle (ATV) approaching a river are shown on the

following Graph. Find the instantaneous velocities at 2.0 s, 3.0 s, and 5.0 s.

A tangent line at 2.0 s is drawn and only

touches the point at 2.0 s. The slope of the

tangent line is the instantaneous velocity at

2.0 s.

At 3.0 s,

At 5.0 s, the tangent is a horizontal line.

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

0.0 1.0 2.0 3.0 4.0 5.0

Po

siti

on

(m

[E]

)

Time (s)

Position vs Time

0.0

10.0

20.0

30.0

40.0

0.0 1.0 2.0 3.0 4.0 5.0

Po

siti

on

(m

[E]

)

Time (s)

Position vs Time

0.0

10.0

20.0

30.0

40.0

0.0 1.0 2.0 3.0 4.0 5.0

Po

siti

on

(m

[E]

)

Time (s)

Position vs Time

0.0

10.0

20.0

30.0

40.0

0.0 1.0 2.0 3.0 4.0 5.0

Po

siti

on

(m

[E]

)

Time (s)

Position vs Time

Tangent Line

Rise ( )

Run ( )

Rise ( )

Run ( )

SPH4C1 Lesson 02 Motion and its Applications

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REVIEW: POSITION-TIME GRAPHS

Positions can be read off the graph.

Displacements are a change in position. Read the initial and final position and subtract.

Average velocity over an interval is the slope of the line joining the end-points of the interval

(the secant).

Instantaneous velocity at a point is slope of the tangent at that point.

When the curve is a parabola, the instantaneous velocity at the centre of an interval is equal

to the average velocity of the entire interval.

PRACTICE

Use the graph below to answer the

following.

A. Fill in the title on the graph

(up or down)

B. Is the skier getting faster or

slower?

C. Determine the average

velocity of the skier…

i) For the first second

ii) From 1.5 s to 3.5 s

iii) From 4 s to 5 s.

iv) Draw a tangent line at 2.0 seconds. Find the slope of this line. What does

this tell you?

0.0

10.0

20.0

30.0

40.0

0.0 1.0 2.0 3.0 4.0 5.0

Po

siti

on

(m

[E]

)

Time (s)

Position vs Time

0.0

10.0

20.0

30.0

40.0

0.0 1.0 2.0 3.0 4.0 5.0

Po

siti

on

(m

[E]

)

Time (s)

Position vs Time

0

5

10

15

20

25

30

35

40

45

0 2 4 6 8

Po

siti

on

(m

)

Time (s)

Position vs Time for a Skier Going _____ a Hill

Rise ( )

Run ( )

Instantaneous Velocity

Rise ( )

Run ( )

Average Velocity

SPH4C1 Lesson 02 Motion and its Applications

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Use the graph at the bottom of the page to answer the following questions.

A. Is the skydiver moving up or down? ___________________________________

B. Is the skydiver getting faster or slower? ___________________________________

C. How far does the skydiver fall in total? ___________________________________

D. How far does the skydiver fall in the first 0.4 seconds?

_________________________________

E. How far does the skydiver fall from 2 s to 2.4 s?

___________________________________

F. Why is your answer from #5 bigger than your answer for #4?

G. Draw tangent lines and find their slope to figure out how fast the skydiver was going at

a. 0.4 s

b. 1.2 s

c. 2.2 s

0

5

10

15

20

25

30

35

0 0.4 0.8 1.2 1.6 2 2.4 2.8

Po

sit

ion

(m

)

Time (s)

Position vs Time for a Skydiver in Free Fall

SPH4C1 Lesson 02 Motion and its Applications

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Use the graph below to answer the following questions.

A. Label all areas as either “speeding up”, “slowing down”, “constant speed” or “stopped”.

Get teacher’s signature when done. ___________

B. At what time did Isay turn around? ______________

Instantaneous Speed: the speed at ONE POINT IN TIME

A. Label each one as an example of average speed or instantaneous speed:

a) speed from a radar gun ________________________

b) slope of position-time graph ________________________

c) total distance divided by total time ________________________

d) speed on a speedometer ________________________

B. How can you find the instantaneous speed of an object from the position-time graph?

C. Draw tangent lines on Isay’s graph at 3.0 s, 10.0 s and 19.0 s. Show your teacher.

_____

D. Find the instantaneous speed at 3.0 s and 19.0 s. Show your work.

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25 30 35

Po

siti

on

(m

)

Time (s)

Position vs. Time for Isay Late for Class

SPH4C1 Lesson 02 Motion and its Applications

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VELOCITY-TIME GRAPHS

Are graphs of instantaneous velocity plotted against time.

Instantaneous velocities can be read from the graph.

Average acceleration over an interval is determined by the slope of the line joining the two

points (the secant).

Instantaneous acceleration at a point is the slope of the tangent at that point.

When the velocity and the acceleration have the same sign, the object is speeding up (the

line goes away from the time axis).

When the velocity and the acceleration have the opposite sign, the object is slowing down

(the line approaches the time axis).

The displacement during an interval can be found from the area between the curve and the

time axis over that interval.

o Areas above the axis are positive and areas below the axis are negative.

Average velocities are found from the definition of average velocity: displacement over time.

INTERPRETING VELOCITY-TIME GRAPHS

Direction of velocity ____________

Velocity is _______________

Value of acceleration is ____________

Direction of velocity ____________

Velocity is _______________

Value of acceleration is ____________

Direction of velocity ____________

Direction of acceleration is ____________

Velocity is _____________ (increasing,decreasing)

0.0

5.0

10.0

15.0

0.0 1.0 2.0 3.0 4.0 5.0

Ve

loci

ty (

m/s

[E]

)

Time (s)

Velocity vs Time

0.0

5.0

10.0

15.0

0 1 2 3 4 5

Ve

loci

ty (

m/s

[E]

)

Time (s)

Velocity vs Time

��[𝑁]

t

��[𝑁]

t

��[𝑁]

t

Rise ( )

Run ( )

Average Acceleration

Displacement – area of coloured shapes under the graph.

SPH4C1 Lesson 02 Motion and its Applications

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Direction of velocity above axis ____________

Direction of velocity below axis ____________

What happened at crossing of axis?

_____________________________

Direction of acceleration is ____________

Velocity above axis is ______________

Velocity below axis is ______________

Direction of velocity ____________

Direction of acceleration is ____________

Velocity is _____________ (increasing,decreasing)

Direction of velocity above axis ____________

Direction of velocity below axis ____________

What happened at crossing of axis?

_____________________________

Direction of acceleration is ____________

Velocity above axis is ______________

Velocity below axis is ______________

POSITION-TIME TO VELOCITY-TIME GRAPHS

You can use the information in a position-time graph to create a velocity-time graph. Find the

slope of each section on the following position-time graph. The sections are determined by a

change in slope and a change in the sign of the position. Make a table of your results below.

Then use the table of values from your position-time graph to plot a velocity time graph.

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t

��[𝑁]

t

��[𝑁]

t

SPH4C1 Lesson 02 Motion and its Applications

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VELOCITY-TIME TO POSITION-TIME GRAPHS

You can use the information in a velocity-time graph to draw a position-time graph. To sketch

the position-time graph, find the area under the velocity-time graph for each section. The

sections are determined by a change in slope or sign of the velocity. Make a table showing the

TOTAL AREA FROM 0 up to each 5 s of time. Then plot the data on the position-time graph.

SPH4C1 Lesson 02 Motion and its Applications

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PRACTICE POSITION-TIME GRAPHS

The position - time graph for an object in motion is shown below.

a) Write a short story that could represent the motion of this graph.

b) What is the displacement in the intervals:

i) AD ii) EI

c) What is the distance traveled in the intervals:

i) AE ii) DH

d) What is the velocity in the intervals:

i) BC ii) FG

e) What segments of the trip have a zero velocity? Explain?

f) What is the average speed in the intervals:

i) AE ii) DH

iii) AH

g) What is the average velocity in the intervals:

i) AC ii) DH

A

B

C

D E

F

G H

I

-6

-4

-2

0

2

4

6

8

10

0 1 2 3 4 5 6 7 8 9Po

siti

on

(m

[N

])

Time (s)

Position-Time Graph

SPH4C1 Lesson 02 Motion and its Applications

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FINDING VELOCITY FROM A CURVED POSITION-TIME GRAPH

1. The ball is on its way up from 0s until _________ and it returns to the ground at

________. On the way up, its speed is _____________________ and you can tell

because the slope is getting ____________________________________. On the way

down, the speed is ____________________ and you can tell because the slope is getting

_____________________. At the maximum height, its speed is _________.

2. Find the average velocity of the ball from 0s until 0.60 s.

3. Find the average velocity of the ball from 1.40 s until 2.2 s.

4. Find the instantaneous velocity of the ball at

a. 0.10 s

b. 1.60 s

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5

Po

sit

ion

(m

[u

p])

Time (s)

Position vs. Time for a Ball Thrown Upward