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Linear Kinematics

Chapter 2 in the text

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KINEMATICSKINEMATICS

LINEARLINEAR ANGULARANGULAR

ScalarsScalarsDistanceDistance

SpeedSpeed

VectorsVectorsDisplacementDisplacement

VelocityVelocityAccelerationAcceleration

Next ClassNext Class

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Scalars

• A measure that only considers magnitude• Does not consider direction• E.g., a distance of 15 meters is a scalar

measure

• The line is 15 meters but has no directionThe line is 15 meters but has no direction

15 m

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Vectors

• Describes both a magnitude and direction• E.g., a displacement of 15 meters in the

positive direction is a vector.• Represented by arrows, in which the

length represents magnitude and orientation represents direction.

+ 15 m

• The arrow is 15 meters in the positive directionThe arrow is 15 meters in the positive direction

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Distance

• A measure of the length of the path followed by an object from its initial to final position.

• A scalar quantity (no direction)

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Speed• The rate of motion of an object• The rate at which an object’s position is

changing.• A scalar quantity (no direction)

Time

Distance SpeedAverage

t

LS

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Displacement

• The straight-line distance in a specific direction from the starting position to the ending position.

• A vector quantity (must have direction)• As the crow flies

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Velocity

• The rate of motion in a specific direction

• Same as speed but with a direction

• A vector quantity

Time

ntDisplacemeVelocity Average

t

DV

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Distance vs. DisplacementNN

WW EE

SS

StartStart

5 km5 km

EndEnd10 km10 km

DistanceDistance

Displacement Displacement

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Speed vs. Velocity

It took Billy 3.5 hours in total to walk 5 km North It took Billy 3.5 hours in total to walk 5 km North and 10 km East. What was Billy’s average speed and 10 km East. What was Billy’s average speed and average velocity?and average velocity?

SpeedSpeed VelocityVelocity

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Acceleration• The rate at which an object’s speed or

velocity changes.• When an object speeds up, slows down,

starts, stops, or changes direction, it is accelerating.

• Always a vector quantity (has direction)

Time

Velocity - VelocityonAccelerati Average initialfinal

t

VVa if

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Acceleration

• The direction of motion does not indicate the direction of acceleration.

• An object can be accelerating even if its speed remains unchanged. The acceleration could be due to a change in direction not magnitude.

Midterm Example

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Bolt runs 200 m in 19.19 seconds. Assume he ran on the inside Bolt runs 200 m in 19.19 seconds. Assume he ran on the inside line of lane 1, which makes a semicircle (r = 36.5 m) for the first line of lane 1, which makes a semicircle (r = 36.5 m) for the first part of the race. He runs the curve in 11 s. part of the race. He runs the curve in 11 s.

36.5 m

1. What distance was run on the curve?

2. What was his displacement after the curve?

3. Total distance?

4. Total displacement?

5. Average velocity on the curve?

6. Average speed on the curve?

7. Average velocity for the race?

8. Average speed for the race?

NN

WW EE

SS

Start

Finish

Circle Circumference = 2r; Circle Diameter = 2r; r is radius

Midterm Example

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Bolt runs 200 m in 19.19 seconds. Assume he ran on the inside line of lane 1, which makes a semicircle (r = 36.5 m) for the first part of the race. He runs the curve in 11 s.

36.5 m

1. What distance was run on the curve?

2. What was his displacement after the curve?

3. Total distance?

4. Total displacement?

5. Average velocity on the curve?

6. Average speed on the curve?

7. Average velocity for the race?

8. Average speed for the race?

NN

WW EE

SS

Start

Finish

Circle Circumference = 2r; Circle Diameter = 2r; r is radius

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Instantaneous Velocity

• The average velocity over an infinitely small time period.

• Determined using Calculus

• The derivative of displacement

• The slope of the displacement curve

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Instantaneous Acceleration

• The average acceleration over an infinitely small time period.

• Determined using Calculus

• The derivative of velocity

• The slope of the velocity curve

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Slope

XX

YY

(0,0)(0,0)

(4,8)(4,8)8

44

Slope = Slope = riserise = = YY = = Y2 – Y1Y2 – Y1 = = 8 – 08 – 0 = = 88 = 2 = 2 runrun X X2 – X1 4 – 0 4 X X2 – X1 4 – 0 4

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Velocity is the slope of Displacement

XX

YY

(0,0)(0,0)

(4,8)(4,8)88

44

AverageAverageVelocity = Velocity = riserise = = DD = = D2 – D1D2 – D1 = = 8 – 08 – 0 = = 8 m8 m = 2 m/s = 2 m/s

runrun t t2 – t1 4 – 0 4 s t t2 – t1 4 – 0 4 s

DisplacementDisplacement(m)(m)

Time (s)Time (s)

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1.1. The displacement graph on the The displacement graph on the previous slide was a straight line, previous slide was a straight line, therefore it’s slope was 2 at every therefore it’s slope was 2 at every instant.instant.

2.2. Which means the velocity at any instant Which means the velocity at any instant is equal to the average velocity.is equal to the average velocity.

3.3. However if the graph was not straight However if the graph was not straight the instantaneous velocity could not be the instantaneous velocity could not be determined from the average velocity.determined from the average velocity.

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Average vs. Instantaneous

AverageAverageVelocity = Velocity = riserise = = DD = = D2 – D1D2 – D1 = = 8 – 08 – 0 = = 8 m8 m = 2 m/s = 2 m/s

run run t t2 – t1 4 – 0 4 s t t2 – t1 4 – 0 4 s

XX

YY

88

44

DisplacementDisplacement(m)(m)

Time (s)Time (s)(0,0)(0,0)

(4,8)(4,8)

The average velocity does not accurately The average velocity does not accurately represent slope at this particular point.represent slope at this particular point.

• Read Ch. 6 pages 147-158 for next class