Chapter 2 Motion in One Dimension 2-1 Displacement and Velocity Motion – takes place over time...

Chapter 2 Motion in One Dimension2-1 Displacement and Velocity

Motion – takes place over time Object’s change in position is relative to a reference point Kinematics = part of physics that describes motion without

discussing the forces that cause the motion

Displacement vs distance traveled Displacement = the length of the straight-lined path

between two points – not the total distance traveled.

Displacement can be positive or negative values During which time intervals did it travel is a positive

direction? What about a negative direction?

Motion in 1 Dimension Use the letter ‘x’ for horizontal motion and“y” for vertical motion (up and down).

Change in position (distance) = ∆X or (∆Y)Greek letter delta (∆) = a change in position

Always (final position ) – (initial position)

Formula for Displacement:

Rise/Run = 600 km/3 hr = 200 km/hr

Rise=?

Slope of a Position vs Time Graph = Speed

3 h

600 km

Speed can interpreted using a Displacement vs Time Graph

Different Slopes

0

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7

Time (hr)

Dis

tan

ce (

km

)

Run = 1 hr

Run = 1 hr

Run = 1 hr

Rise = 0 km

Rise = 2 km

Rise = 1 km

Slope = Rise/Run= 1 km/1 hr= 1 km/hr

Slope = Rise/Run= 0 km/1 hr= 0 km/hr

Slope = Rise/Run= 2 km/1 hr= 2 km/hr

Difference between Velocity and Speed Velocity describes motion with both a direction

and a numerical value (a magnitude). Moving at 65 mph due North Speed has no direction, only magnitude.

distance traveledaverage speed =

time of travel

Formula:V = d/t (direction)S = d/t

Average velocity is the total displacement divided by the time interval during which the displacement occurred.

Vavg = Vi + Vf 2

And if Vi = 0, then, Vavg = ½ vf

10

9

D 8is 7pl 6ac 5em 4en 3t

2(m)

1

0 1 2 3 4

Time (s)

Finding Average Velocity if Velocity is not constant

10

9

D 8is 7pl 6ac 5em 4en 3t

2(m)

1

0 1 2 3 4

Time (s)

*

*

Finding Average Velocity- pick 2 points

10

9

D 8is 7pl 6ac 5em 4en 3t

2(m)

1

0 1 2 3 4

Time (s)

*

*

Finding Average Velocity- draw a line between them

10

9

D 8is 7 (3.5 , 7)pl 6ac 5em 4en 3t

2(m) (0 , 0)

1

0 1 2 3 4

Time (s)

*

*

Finding Average Velocity- find their coordinates

10

9

D 8is 7 (3.5 , 7)pl 6ac 5em 4en 3t

2(m) (0 , 0)

1

0 1 2 3 4

Time (s)

*

*

Finding Average Velocity- calculate the slope

∆x xf - xi

∆t tf - ti

10

9

D 8is 7 (3.5 , 7)pl 6ac 5em 4en 3t

2(m) (0 , 0)

1

0 1 2 3 4

Time (s)

*

*

∆x xf - xi 7 - 0

∆t tf - ti 3.5 - 0

Finding Average Velocity- calculate the slope

10

9

D 8i m/ss 7 (3.5 , 7)pl 6ac 5em 4en 3t

2(m) (0 , 0)

1

0 1 2 3 4

Time (s)

*

*

∆x xf - xi 7 - 0

∆t tf - ti 3.5 - 02

Finding Average Velocity- calculate the slope

We can calculate the velocity of a moving object at any point along the curve.

This is called the -

Instantaneous Velocity

We can calculate the velocity of a moving object at any point along the curve.

This is called the -

Instantaneous VelocityDraw a line tangent to the velocity curve, and find its slope –

∆xtan x2 - x1

∆ttan t2 - t1Vinst

Speedometer

10

9

D 8is 7pl 6ac 5em 4en 3t

2(m)

1

0 1 2 3 4

Time (s)

.

10

9

D 8is 7pl 6ac 5em 4en 3t

2(m)

1

0 1 2 3 4

Time (s)

Finding Instantaneous Velocity- draw the tangent line

10

9

D 8is 7pl 6ac 5em 4en 3t

2(m)

1

0 1 2 3 4

Time (s)

*

*

Finding Instantaneous Velocity- find 2 convenient points

10

9

D 8 i (4 , 7)s 7pl 6ac 5em 4en 3t

2(m)

1 (2 , 1)

0 1 2 3 4

Time (s)

*

*

Finding Instantaneous Velocity- find their coordinates

10

9

D 8 i (4 , 7)s 7pl 6ac 5em 4en 3t

2(m)

1 (2 , 1)

0 1 2 3 4

Time (s)

*

*

Finding Instantaneous Velocity- calculate the slope

∆x x2 - x1 ∆t t2 - t1

10

9

D 8 i (4 , 7)s 7pl 6ac 5em 4en 3t

2(m)

1 (2 , 1)

0 1 2 3 4

Time (s)

*

*

Finding Instantaneous Velocity- calculate the slope

∆x x2 - x1 7 - 1 ∆t t2 - t1 4 - 2

10

9

D 8 i m/s (4 , 7)s 7pl 6ac 5em 4en 3t

2(m)

1 (2 , 1)

0 1 2 3 4

Time (s)

*

*

Finding Instantaneous Velocity- calculate the slope

∆x x2 - x1 7 - 1 ∆t t2 - t1 4 - 2

3

2.2 ACCELERATION – The change in velocity over time.

In Physics we use the expression:

∆v vf - vi

∆t tf - ti

The units for acceleration are usually

meters ( m/s/s ) or m/s2

seconds2

a = =

Velocity -Time Graphs• Slope on a velocity time graph is acceleration.

Time (s)

Vel

ocity

(m

/s)

Slope = ___________

Acceleration = _______________

Rise (ΔY)

Run (Δ X)

Δ Velocity (ΔY)

Δ Time (Δ X)(ΔY)

(Δ X)

Therefore: slope of V-T graph = acceleration

1. What is the final velocity of a car that accelerates from rest at 4 m/s/s for three seconds

2. 2. What is the slope of the line for the red car for the first three seconds? 3. Does the red car pass the blue car at three seconds? If not, then when does the red car pass the blue car? 4. When lines on a velocity-time graph intersect, does it mean that the two cars are passing by each other? If not, what does it mean?

1. What is the final velocity of a car that accelerates from rest at 4 m/s/s for three seconds? 12 m/s2. What is the slope of the line for the red car for the first three seconds? 4 m/s2

3. Does the red car pass the blue car at three seconds? If not, then when does the red car pass the blue car? No, at 9 sec4. When lines on a velocity-time graph intersect, does it mean that the two cars are passing by each other? If not, what does it mean? No, just same velocity

Slope of a velocity vs time = acceleration rise/ run = ∆v/∆t = acceleration

IMPORTANT FORMULAS: Displacement and Final Velocity

For an object that accelerates from rest (vi = 0)

∆x = ½ ( vf ) ∆t ( remember: ½ vf = vavg )

vf = a ( ∆t )

∆x = ½ a( ∆t )2

Vf2 = 2(a)(∆x)

Practice Graph MatchingDraw a position versus time graph for each of the following:

• constant forward motion• constant backward motion• constant acceleration• constant deceleration• sitting still

Time (s)P

ositi

on (

m)

constant forward motion

Straight sloped line going higher [slope (therefore velocity) does not change]

Time (s)

Pos

ition

(m

)

constant backward motion

Straight sloped line going lower [slope (therefore velocity) does not change]

Time (s)

Pos

ition

(m

)

constant acceleration

Time (s)

Pos

ition

(m

)

Increasing slope[slope (therefore velocity) increases]

constant deceleration

Time (s)

Pos

ition

(m

)

Decreasing slope[slope (therefore velocity) decreases]

sitting still

Straight line with no slope

Time (s)

Pos

ition

(m

)

Draw a velocity versus time graph for each of the following:

• constant forward motion• constant backward motion• constant acceleration• constant deceleration• sitting still

constant forward motion

• Velocity stays the same (above 0 m/s)

Time (s)

Vel

ocity

(m

/s)

constant backward motion

• Velocity stays the same (below 0 m/s)

Time (s)

Vel

ocity

(m

/s)

constant acceleration

• Constant upwards slope• Velocity at the second point is more than the

first

Time (s)

Vel

ocity

(m

/s)

constant deceleration

• Constant downward slope• Velocity at the second point is less than the

first

Time (s)

Vel

ocity

(m

/s)

sitting still

• Flat line at 0 velocity

Time (s)

Vel

ocity

(m

/s)