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Chapter 2 Motion in One Dimension 2-1 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.

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

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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:

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3 h 600 km Speed can interpreted using a Displacement vs Time Graph

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Different Slopes 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

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

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Average velocity is the total displacement divided by the time interval during which the displacement occurred. V avg = V i + V f 2 And if V i = 0, then, V avg = ½ v f

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10 9 D 8 i s 7 p l 6 a c 5 e m 4 e n 3 t 2 (m) 1 0 1 2 3 4 Time (s) Finding Average Velocity if Velocity is not constant

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10 9 D 8 i s 7 p l 6 a c 5 e m 4 e n 3 t 2 (m) 1 0 1 2 3 4 Time (s) * * Finding Average Velocity - pick 2 points

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10 9 D 8 i s 7 p l 6 a c 5 e m 4 e n 3 t 2 (m) 1 0 1 2 3 4 Time (s) * * Finding Average Velocity - draw a line between them

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10 9 D 8 i s 7 (3.5, 7) p l 6 a c 5 e m 4 e n 3 t 2 (m) (0, 0) 1 0 1 2 3 4 Time (s) * * Finding Average Velocity - find their coordinates

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10 9 D 8 i s 7 (3.5, 7) p l 6 a c 5 e m 4 e n 3 t 2 (m) (0, 0) 1 0 1 2 3 4 Time (s) * * Finding Average Velocity - calculate the slope ∆x x f - x i ∆t t f - t i

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10 9 D 8 i s 7 (3.5, 7) p l 6 a c 5 e m 4 e n 3 t 2 (m) (0, 0) 1 0 1 2 3 4 Time (s) * * ∆x x f - x i 7 - 0 ∆t t f - t i 3.5 - 0 Finding Average Velocity - calculate the slope

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10 9 D 8 i m/s s 7 (3.5, 7) p l 6 a c 5 e m 4 e n 3 t 2 (m) (0, 0) 1 0 1 2 3 4 Time (s) * * ∆x x f - x i 7 - 0 ∆t t f - t i 3.5 - 0 2 Finding Average Velocity - calculate the slope

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We can calculate the velocity of a moving object at any point along the curve. This is called the - Instantaneous Velocity

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We can calculate the velocity of a moving object at any point along the curve. This is called the - Instantaneous Velocity Draw a line tangent to the velocity curve, and find its slope – ∆x tan x 2 - x 1 ∆t tan t 2 - t 1 V inst Speedometer

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10 9 D 8 i s 7 p l 6 a c 5 e m 4 e n 3 t 2 (m) 1 0 1 2 3 4 Time (s).

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10 9 D 8 i s 7 p l 6 a c 5 e m 4 e n 3 t 2 (m) 1 0 1 2 3 4 Time (s) Finding Instantaneous Velocity - draw the tangent line

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10 9 D 8 i s 7 p l 6 a c 5 e m 4 e n 3 t 2 (m) 1 0 1 2 3 4 Time (s) * * Finding Instantaneous Velocity - find 2 convenient points

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10 9 D 8 i (4, 7) s 7 p l 6 a c 5 e m 4 e n 3 t 2 (m) 1 (2, 1) 0 1 2 3 4 Time (s) * * Finding Instantaneous Velocity - find their coordinates

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10 9 D 8 i (4, 7) s 7 p l 6 a c 5 e m 4 e n 3 t 2 (m) 1 (2, 1) 0 1 2 3 4 Time (s) * * Finding Instantaneous Velocity - calculate the slope ∆x x 2 - x 1 ∆t t 2 - t 1

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10 9 D 8 i (4, 7) s 7 p l 6 a c 5 e m 4 e n 3 t 2 (m) 1 (2, 1) 0 1 2 3 4 Time (s) * * Finding Instantaneous Velocity - calculate the slope ∆x x 2 - x 1 7 - 1 ∆t t 2 - t 1 4 - 2

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10 9 D 8 i m/s (4, 7) s 7 p l 6 a c 5 e m 4 e n 3 t 2 (m) 1 (2, 1) 0 1 2 3 4 Time (s) * * Finding Instantaneous Velocity - calculate the slope ∆x x 2 - x 1 7 - 1 ∆t t 2 - t 1 4 - 2 3

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2.2 ACCELERATION – The change in velocity over time. In Physics we use the expression: ∆v v f - v i ∆t t f - t i The units for acceleration are usually meters ( m/s/s ) or m/s 2 seconds 2 a ==

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Velocity -Time Graphs Slope on a velocity time graph is acceleration. Time (s) Velocity (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

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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?

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1. What is the final velocity of a car that accelerates from rest at 4 m/s/s for three seconds? 12 m/s 2. What is the slope of the line for the red car for the first three seconds? 4 m/s 2 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 sec 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? No, just same velocity

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

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IMPORTANT FORMULAS: Displacement and Final Velocity For an object that accelerates from rest (v i = 0) ∆x = ½ ( v f ) ∆t ( remember: ½ v f = v avg ) v f = a ( ∆t ) ∆x = ½ a( ∆t ) 2 V f 2 = 2(a)(∆x)

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Practice Graph Matching Draw a position versus time graph for each of the following: constant forward motion constant backward motion constant acceleration constant deceleration sitting still Time (s) Position (m)

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constant forward motion Straight sloped line going higher [slope (therefore velocity) does not change] Time (s) Position (m)

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constant backward motion Straight sloped line going lower [slope (therefore velocity) does not change] Time (s) Position (m)

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constant acceleration Time (s) Position (m) Increasing slope [slope (therefore velocity) increases]

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constant deceleration Time (s) Position (m) Decreasing slope [slope (therefore velocity) decreases]

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sitting still Straight line with no slope Time (s) Position (m)

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Draw a velocity versus time graph for each of the following: constant forward motion constant backward motion constant acceleration constant deceleration sitting still

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constant forward motion Velocity stays the same (above 0 m/s) Time (s) Velocity (m/s)

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constant backward motion Velocity stays the same (below 0 m/s) Time (s) Velocity (m/s)

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constant acceleration Constant upwards slope Velocity at the second point is more than the first Time (s) Velocity (m/s)

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constant deceleration Constant downward slope Velocity at the second point is less than the first Time (s) Velocity (m/s)

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sitting still Flat line at 0 velocity Time (s) Velocity (m/s)

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