Physics 2ALecture 2A

"You must learn from the mistakes of others. You can't possibly live long enough to make them all

yourself."--Sam Levenson

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Motion

Chapter 2 will focus on motion in one dimension.

Any description of motion involves three concepts:

1) Displacement

2) Velocity

3) Acceleration

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DisplacementNearly the first thing you must do in every Physics problem is define a coordinate system.

Choosing the proper coordinate system can make a huge difference.

For one dimensional motion it is rather easy.

1) Choose an origin.

2) Choose a positive direction.

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Displacementx is defined as the position compared to the origin.

Displacement, ∆x, is a difference between positions.

∆x = x2–x1

Distance, d, is the total length travelled.

d can only be a positive value.

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2-D VariablesIf we try to describe motion in two dimensions we turn to the vector, r.

Position will now be given by:

Displacement will be given by:

Note that position depends on where you choose the origin while displacement does not.

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VelocityThe velocity of an object is its displacement over a period of time.

Average velocity, vavg, is:

The speed of an object is its distance travelled over a period of time.

Average speed is:

Average speed is only a positive value.

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VelocityGraphically, we find average velocity by examining the rise (Δx) over the run (Δt) in an x vs. t graphBetween points A and B, we find that the average velocity would be:

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VelocityInstantaneous velocity is the velocity at a given instant of time.

Instantaneous velocity, v, is:

Instantaneous speed will just be the magnitude of the instantaneous velocity (only positive).

We usually “drop” instantaneous when talking about velocity or speed.

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Velocity

It is different from average velocity in that you do not care what happens over a time period only a time instant.

Graphically, we find instantaneous velocity by examining the slope of an x vs. t graph at a particular point.

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VelocityExampleRunner A is initially 4.0 miles west of a flagpole and is running with a constant velocity of 6.0 mph due east. Runner B is initially 3.0 miles east of the flagpole and is running with a constant velocity of 5.0 mph due west. How far are the runners from the flagpole when they meet?

AnswerFirst, you must define a coordinate system.Let’s choose the flagpole as the origin with east being the positive x-direction.

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VelocityAnswerEach runner needs their own separate variables measured from the flagpole:For Runner A =>

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voA =ΔxAt

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ΔxA = voA( )t

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ΔxB = voB( )tSimilarly for Runner B =>

But recall that displacement is:

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ΔxB = xB , final − xB ,initial

We can thus rewrite our equations for Runner A and B as:

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xA , final = xA ,initial + voA( )t

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xB , final = xB ,initial + voB( )t

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VelocityAnswerWhen the runners meet their positions will be equal:

We know every variable here except for time, so we can plug them in and solve for time.

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xB , final = xA , final

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xB ,initial + voB( )t = xA ,initial + voA( )t

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3.0 miles + −5.0 mihr( )t = −4.0 miles + 6.0 mi

hr( )t

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7.0 miles = 11.0 mihr( )t

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t =7.0

11.0 hr = 0.636 hr <= Yeah we’re

done! Nope!12

VelocityAnswerWe wanted to know how far from the flagpole the runners meet.We can use either position equation to solve:

where the negative sign means that they meet to the west of the flagpole.

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xB , final = xB ,initial + voB( )t = 3.0 miles + −5.0 mihr( ) 0.636 hr( )

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xB , final = −0.18 miles

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2-D VariablesIn two dimensions, average velocity is again the total displacement over the time interval:

Since Δt is a scalar and always positive, average velocity will always point in the direction of displacement.

Instantaneous velocity is again the velocity of an object at any instant of time.

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v = limΔt→0

Δ r Δt

=d r dt

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AccelerationThe acceleration of an object is how much its velocity changes over a period of time.

Average acceleration, aavg, is:

Instantaneous acceleration is the acceleration of an object at a given instant of time.

Acceleration is:

Acceleration is usually measured in units of m/s2.

But you can also find it in other units so be careful.

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AccelerationWatch out for the sign of acceleration.

Positive acceleration does not always mean “speeding up.”

Nor does negative acceleration always mean “slowing down.”

When acceleration and velocity are in the same direction, the object “speeds up.”

When acceleration and velocity are in opposite directions, the object “slows down.”

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2-D VariablesIn two dimensions, average acceleration is again the change in velocity over the time interval:

Since Δt is a scalar and always positive, average acceleration will always point in the direction of the change of velocity.

Instantaneous acceleration is again the acceleration of an object at any instant of time.

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a = limΔt→0

Δ v Δt

=d v dt

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For Next Time (FNT)

Keep working on the homework for Chapter 2.

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