chapter 2 motion in one dimension 2-1 displacement and velocity motion – takes place over time...
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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)