College and Engineering Physics

Velocity and Speed

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Motion in One Dimension

Uniform Motion

Instantaneous Velocity

Finding Position from Velocity

The Particle Model

Velocity

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A particle is an object that can be represented as a mass at a single point in space.

This point is called the center of mass of the object. We will learn about this later.

The relationship between the force on the particle and the particle’s acceleration determine the path (trajectory) that the particle will follow.

The Particle Model

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Velocity measures the motion of an objectIt consists of two parts …

1. The magnitude (speed) which is larger when the object is moving fastand smaller when the object is moving slow.

The turtle has a smaller magnitude of velocity than the rabbit.

2. The direction tells you where the object is going.

The velocity of the airplanes has the same magnitude,but a different direction.

Velocity v

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The x-component of most vectors (though not all) will be the component which is horizontal (lies along the plane of the earth).

Horizontal Motion

PositionDetermines a

particle’s location

VelocityDetermines the direction of the

particle’s motion

Acceleration

Determines the direction of the

change in a particle’s motion

x

xv

xa

Variable Symbol NegativePositive

Meaningof theSign

0 0

y

x

y

x

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The y-component of most vectors (though not all) will be the component which is vertical (points perpendicular to the plane of the earth).

Vertical Motion

PositionDetermines a

particle’s location

VelocityDetermine the direction of the

particle’s motion

Acceleration

Determines the direction of the

change in a particle’s motion

y

yv

ya

Variable Symbol NegativePositive

Meaningof theSign

0 0

y

x

y

x

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The direction and quantity of “motion” of a particle is given by its velocity.

The average velocity of a particle (whether or not the motion is uniform) is given be a simple equation …

In words, we would say this equation tells us that “the average velocity of a particle is its displacement over the interval during which it moved”.

The slope of this line is the x-component of the

average velocity.

Average Velocity

t

rvavg

x

t

t

x

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Velocity is constant if it does not change.

This means that the quantity (known as the magnitude) and the direction stay the same.

If the position vs. time graph of a particle’s motion is a straight line,

then the average velocity is the slope of this line and this component of

the motion is uniform.

Motion with Constant Velocity (Uniform Motion)

x

t

t

x

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To model the way a particle moves, we can use a graph of position vs. time.

Position vs. Time Graph

x

ty

t

y

x

tr vs.

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Another way to model the motion of a particle is to graph velocity vs. time.

Velocity is the derivative (slope) of the position over time.

Velocity vs. Time Graph

x

t

vx

t

0s 5s 8s4s 7s2s 3s 6s1s 0s 5s 8s4s 7s2s 3s 6s1s

tv vs.

drv

dt

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4

-1

-4

0

-3

2

1

-2

3

To determine where an object will be at a later time, we can use velocity vs. time curve.

Velocity vs. Time vs. v t

Vx (m/s)

t (s)

0 5 84 72 3 61

Time Velocity Position

0 s 3 m/s 2 m

1s 3 m/s 5 m

2 s 3 m/s 8 m

3 s -3 m/s 5 m

4 s -3 m/s 2 m

5 s -3 m/s -1 m

6 s 0 m/s -1 m

7 s 0 m/s -1 m

8 s 2 m/s 1 m0

0

t

xtx x v dt

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8

-2

-8

0

-6

4

2

-4

6

To determine where an object will be at a later time, we can use velocity vs. time curve.

Position vs. Time vs. x t

x (m)

t (s)

0 5 84 72 3 61

Time Velocity Position

0 s 3 m/s 2 m

1s 3 m/s 5 m

2 s 3 m/s 8 m

3 s -3 m/s 5 m

4 s -3 m/s 2 m

5 s -3 m/s -1 m

6 s 0 m/s -1 m

7 s 0 m/s -1 m

8 s 2 m/s 1 m0

0

t

xtx x v dt

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8

-2

-8

0

-6

4

2

-4

6

To determine where an object will be at a later time, we can use velocity vs. time curve.

Comparison

x (m)

t (s)

0 5 84 72 3 61

4

-1

-4

0

-3

2

1

-2

3

Vx (m/s)

0 5 84 72 3 61

0 2 mx 0 3 mx 0

0

t

xtx x v dt

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