Representing Motion. Motion We are looking to ____________and ____________an object in motion. Three “rules” we will follow: –The motion is in a __________________.

Representing Motion

Motion

• We are looking to ____________and ____________an object in motion.

• Three “rules” we will follow:

– The motion is in a __________________

– The _________of the motion is ignored (coming soon!)

– The objects considered is a __________(not for long!)

• Particles and particle like objects move uniformly

– Ex. __________________

– ANTI Ex. _______________________

Position• The ____________of the particle in space.

• Needs a ______________description to be useful.

• We assign a ___________to represent the particles position on a coordinate grid.– There needs to be a ______point to reference– The positions to the left are ___________– The positions to the right are ____________

Distance

• _________: The total path length when moving from one location to another– __________

• Has only magnitude (i.e. size) and unit (NO DIRECTION)

• Other scalars: mass, time, energy

Displacement

• Displacement: the straight-line ____________between two points, along with the ____________from the starting point to the finish point– Vector: Has a ____________, unit, and

direction• Other vectors: acceleration, momentum, etc.• Ex. 23 m/s [W] or -23 m/s

Displacement cont’d

• Position: x (m)

• Displacement: ______________________

+x→ ← -x

+x→ ← -x

Displacement vs. Distance

+x→ ← -x

+x→ ← -x

Speed

• Speed: ______which distance is traveled– Average Speed ( ): The distance, d,

traveled over the total time of the trip

_______________________

– Instantaneous speed: speed at a particular instant

s

distance that travel to timetotal

traveleddistance speed average

t

ds

Speed cont’d

Instantaneous or Average Speed?

Speed cont’d• Example 1:

A car is moving at a constant speed. If the car traveled a distance of 60 meters in 4 seconds:

a) Find the speed of the car

At the 60 meter mark, the car suddenly slows down to rest over 3.5 seconds and covers another 15 meters in doing so.

b) Find the average speed of the car over the course of the entire problem

Velocity

• Velocity: how fast something is moving and in which ____________

___________________

• Direction of velocity is determined by the direction of the _______________

time traveltotal

ntdisplaceme velocityaverage

t

xv

Instantaneous Velocity

• Instantaneous velocity: How fast something is moving at a ___________time (w/ a direction)

• Defined later as the ________of a position-time (x vs t) graph

Graphing Motion

Carl Lewis ‘88 Usain Bolt ‘080-10 m 1.89 s 1.85 s

10-20 m 2.96 s 2.87 s

20-30 m 3.90 s 3.78 s

30-40 m 4.79 s 4.65. s

40-50 m 5.65 s 5.50 s

50-60 m 6.48 s 6.32 s

60-70 m 7.33 s 7.14 s

70-80 m 8.18 s 7.96 s

80-90 m 9.04 s 8.79 s

90-100 m 9.92 s 9.69 s

Bolt ‘08 vs. Lewis ‘88

Position (x) vs Time Graphs

30 m 40 m 50 m 60 m20 m10 m0 m

3 s 4 s 5 s 6 s2 s1 s0 sX

(m)

t(s)

0 m

0 s 6 s

60 m0

1

2

3

4

5

6

0

10

20

30

40

50

60

x(m)t(s)

Avg Velocity vs Instantaneous Velocity

• On a Position vs. Time graph:– Avg _________is the displacement divided by time

interval over which it occurred– Instantaneous velocity is the ______of a line at a

given point• If the slope is constant along a line segment Avg. Velocity =

Inst. Velocity

• If the slope is changing vinst = slope of a line ___________at a given point.

X(m

)

t(s) 5

1. position at t = 7 s 

2. distance from 2 to 4 s

3. distance from 0 to 5 s

4. displacement from 0 to 4 s

5. displacement from 0 to 3 s

6. Time interval where speed is changing

7. speed at t = 2.5 s

8. instantaneous speed at t = 4 s

9. average velocity from 3 to 7 s

10. average speed from 0 to 8 s

Graphing Summary up to now

x(m)

t(s)

Straight line means NO acceleration

x(m)

t(s)

Curved line means changing slope which means changing ____________

Interpreting a VvT Graphv(

m/s

)

t(s)

+, ___________velocity

v(m

/s)

t(s)

+, ___________velocity v(m

/s)

t(s)

+, ___________velocityv(

m/s

)

t(s)

-, ___________velocity v(m

/s)

t(s)-, ___________velocity

Interpreting a VvT Graphv(

m/s

)

t(s)

v(m

/s)

t(s)

v(m

/s)

t(s)

v(m

/s)

t(s)

If the line is in the positive (+) portion of the graph the object is moving ________(i.e., + direction)

If the line is in the negative (-) portion of the graph the object is moving ________(i.e., - direction)

Interpreting a VvT Graphv(

m/s

)

t(s)

v(m

/s)

t(s)

If the line passes from the one region (+ to – or – to +) to another, the object changes _______________

Time (s)

v (m

/s)

Interpreting a Velocity vs. Time Graph

The area under the curve is the objects _____________.

Interpreting a Velocity vs. Time Graph

The area under the curve is the objects displacement.

Time (s)

v (m

/s)

Interpreting a Velocity vs. Time Graph

The area under the curve is the objects displacement.

Time (s)

v (m

/s)

velocity-time graph

-4-3-2-101234567

0 1 2 3 4 5 6 7 8 9 10

t (s)

v (m

/s)

• What is the displacement from 0s to 2s?

• What is the displacement from 6s to 7 s?

• What is the displacement from 8s to 10s?

velocity-time graph 2

0

5

10

15

20

25

0 10 20 30 40 50 60

t (s)

v (m

/s)

• What is the total displacement?

Interpreting a VvT Graph

The slope of a line on a VvT graph indicates acceleration (unit: m/s2).

__________________

_____________________

Time (s)

v (m

/s)

v

t

Acceleration

• Acceleration: The time rate of change of velocity

Vector (has magnitude and direction)

Unit: __________

The slope of a line on a velocity vs. time graph

Visualizing Acceleration

Practice Problem

The United States Bowling Congress conducted a study on ideal bowling ball speed. It was found that a bowling ball should leave the hand going 9.4 m/s. If the ball goes from being at rest to 9.4 m/s in 1.5 seconds what is the acceleration of the ball?

Practice Problem

• A man starts from rest and is accelerated at a rate of 453.6 m/s2 over a time interval of .75s. What was his final velocity?

Acceleration Expressed in g’s

– When accelerations are _________we express them as a multiple of “g”

•

• It is the ___________due to gravity near the surface of the Earth

Constant Acceleration

• This is a special case that tends to simplify things.

• _______________, or mostly constant, acceleration occurs all the time.– Car starting from rest when a light turns green

– Car braking at a light when a light turns red

• There are a set of _________that are used to describe this motion.

Kinematic Equations

Constant Acceleration Problem

• A car starts from rest and accelerates uniformly to 23 m/s in 8 seconds. What distance did the car cover in this time?

Graphical Look at Motion: displacement – time curve

• The _______of the curve is the velocity

• The curved line indicates the __________is changing– Therefore, there is an

_______________

Graphical Look at Motion: velocity – time curve

• The slope gives the ___________

• The straight line indicates a __________acceleration

• The zero slope indicates a ___________acceleration

Graphical Look at Motion: acceleration – time curve

Test Graphical Interpretations

• Match a given velocity graph with the corresponding acceleration graph

Free Fall Acceleration

• This is a case of constant acceleration that occurs ___________.

• All things fall to the Earth with the same ________________________________–In the absence of air ________________, all things fall to the Earth with the same acceleration:

–This is invariant of the objects dimensions, density, weight etc.

• When using the kinematic equations we use–ay = -g = -9.80 m/s2

Free Fall – an object dropped

• Initial velocity is zero• Let up be positive• Use the kinematic

equations– Generally use y instead

of x since vertical

• Acceleration is – ay = -g = -9.80 m/s2

_____

______

Free Fall – an object thrown downward

• ay = -g = -9.80 m/s2

• Initial velocity 0– With __________being

positive, initial velocity will be negative

______

______

Free Fall -- object thrown upward

• Initial velocity is upward, so ____________

• The instantaneous velocity at the _____________height is zero

• ay = _______________everywhere in the motion

v = 0

______

______

Thrown upward, cont.

• The motion may be __________– Then tup = tdown

– Then v = -vo

• The motion may not be symmetric– Break the motion into various parts

• Generally up and down

Free Fall Example

• Initial velocity at A is upward (+) and acceleration is ______________

• At B, the velocity is 0 and the is ________________

• At C, the ______________has the same magnitude as at A, but is in the opposite direction

• The ________________is –50.0 m (it ends up 50.0 m below its starting point)

Vertical motion sample problem

• A ball is thrown upward with an initial velocity of 20 m/s.– What is the max height the ball will reach?

– What will the velocity of the ball be half way to the maximum height?

– What will the velocity of the ball be half way down to the hand?

– What is the total time the ball is in the air?