PreClass Notes: Chapter 2 - U of T Physicsjharlow/teaching/phy131f15/cl03v...2015-06-16 2 © 2012 Pearson Education, Inc. Slide 1-3 Position and Displacement •In one dimension, position

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© 2012 Pearson Education, Inc. Slide 1-1

PreClass Notes: Chapter 2

• From Essential University Physics 3rd Edition

• by Richard Wolfson, Middlebury College

• ©2016 by Pearson Education, Inc.

• Narration and extra little notes by Jason Harlow,

University of Toronto

• This video is meant for University of Toronto

students taking PHY131.


Outline

“The study of motion without

regard to its cause is called

kinematics” – R.Wolfson

• 2.1 Average Velocity

• 2.2 Instantaneous Velocity

• 2.3 Acceleration

• 2.4 Constant Acceleration

• 2.5 Acceleration due to

Gravity, Freefall

• 2.6 Non-constant

Acceleration

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Position and Displacement

• In one dimension, position can be described by a

positive or negative number on a number line, also

called a coordinate system.

– Position zero, the origin of the coordinate system, is

arbitrary and you’re free to choose it wherever it’s

convenient.

• Displacement is change in position.

– For motion along the x direction, displacement is

designated ∆x:

∆x = x2–x1

where x1 and x2 are the initial and final positions,

respectively.


Constant Velocity

For uniform motion, the

position-versus-time

graph is a straight line

The average velocity is

the slope of the position-

versus-time graph

The SI units of velocity

are m/s.

𝑣 =∆𝑥

∆𝑡= slope of the position vs. time graph

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Vocabulary Review…

The distance an object travels is a scalar quantity (no

direction given, always positive)

The displacement of an object is a vector quantity,

equal to the final position minus the initial position

An object’s speed v is scalar quantity (no direction

given, always positive)

Velocity is a vector quantity that includes direction

In one dimension, the direction of velocity is specified

by the + or − sign


Example 1: You drive from

International Falls to Des Moines in

10 hr. What was your displacement

and average velocity?

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GOT IT?

• When an object moves from one point in space to

another, the magnitude of its displacement is

A. either less than or equal to the distance traveled.

B. either greater than or equal to the distance traveled.

C. either greater than or smaller than the distance

traveled.

D. always equal to the distance traveled.

E. either greater than, smaller than, or equal to the

distance traveled.


Instantaneous Velocity

An object that is speeding up or slowing down is not in

uniform motion

In this case, the position-versus-time graph is not a straight

line

We can determine the average speed 𝑣 between any two

times separated by time interval Δt by finding the slope of the

straight-line connection between the two points

The instantaneous velocity is the is the object’s velocity at a

single instant of time t

The average velocity 𝑣 = Δx/Δt becomes a better and better

approximation to the instantaneous velocity as Δt gets smaller

and smaller

𝑣 = lim∆𝑡→0

∆𝑥

∆𝑡=𝑑𝑥

𝑑𝑡

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Instantaneous Velocity

The instantaneous velocity at time t is the average velocity

during a time interval Δt centered on t, as Δt approaches zero

In calculus, this is called the derivative of x with respect to t

Graphically, Δx/Δt is the slope of a straight line

In the limit Δt 0, the straight line is tangent to the curve

The instantaneous velocity at time t is the slope of the line

that is tangent to the position-versus-time graph at time t

v = the slope of the position-versus-time graph at t


GOT IT?

• The figure shows position-versus-time graphs for four

objects. Which starts slowly and then speeds up?

A. B. C. D.

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Using Calculus to Find Derivatives

• In calculus, the derivative gives the result of the

limiting procedure.

– Derivatives of powers are straightforward:

– Other common derivatives include the trig functions:


Acceleration

• Sometimes an object’s velocity changes as it moves.

• Acceleration describes a change in velocity.

• Consider an object whose velocity changes from to

during the time interval ∆t.

• The quantity is the change in velocity.

• The rate of change of velocity is called the average

acceleration:

The Yamaha VMAX accelerates

from 0 to 60 mph in 2.5 s.

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Acceleration

Imagine a competition between a

Volkswagen Beetle and a Porsche

to see which can achieve a velocity

of 30 m/s in the shortest time

The table shows the velocity of

each car, and the figure shows the

velocity-versus-time graphs

Both cars achieved every

velocity between 0 and 30 m/s, so

neither is faster

But for the Porsche, the rate at

which the velocity changed was ∆𝑣

∆𝑡=30 m/s

6.0 s= 5.0 (m/s)/s


Motion with Constant Acceleration

The SI units of acceleration are (m/s)/s, or m/s2

It is the rate of change of velocity, and measures how

quickly or slowly an object’s velocity changes

The average acceleration during a time interval Δt is

Graphically, 𝑎 is the slope of a straight-line velocity-versus-

time graph

If acceleration is constant, the acceleration a is the same as

𝑎.

Acceleration, like velocity, is a vector quantity and has both

magnitude and direction

𝑎 =∆𝑣

∆𝑡

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GOT IT?

• What is the slope of a line connecting two points

on a velocity-versus-time graph?

A. Instantaneous velocity

B. Average velocity

C. Average acceleration

D. Instantaneous acceleration

E. None of the above


Chapter 2 Big Idea

• From the Chapter 2 Summary on page 27:

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Motion with constant velocity Motion with constant acceleration

t

x

t

x

t

v

t

v

t

a

t

a


Constant Acceleration

• When acceleration is constant, then position,

velocity, acceleration, and time are related by

where x0 and v0 are initial values at time t = 0, and x

and v are the values at an arbitrary time t.

0

10 02

210 0 2

2 2

0 02

v v at

x x v v t

x x v t at

v v a x x

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2.5 The Acceleration of Gravity

The motion of an object moving

under the influence of gravity only,

and no other forces, is called free

fall

Two objects dropped from the

same height will, if air resistance

can be neglected, hit the ground at

the same time and with the same

speed

Consequently, any two objects in

free fall, regardless of their mass,

have the same acceleration:

The apple and feather seen

here are falling in a vacuum.


The Acceleration of Gravity

The velocity graph is a

straight line with a slope:

where g is a positive

number which is equal to

9.80 m/s2 on the surface

of the earth

Other planets have

different values of g

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The Acceleration of Gravity

• The equations for constant acceleration apply for

free fall.

– In a coordinate system with y axis upward, they

read:

0

10 02

210 0 2

2 2

0 02

v v gt

y y v v t

y y v t gt

v v g y y


2.6 Non-Constant Acceleration

Figure (a) shows a realistic

velocity-versus-time graph for a car

leaving a stop sign

The graph is not a straight line, so

this is not motion with a constant

acceleration

Figure (b) shows the car’s

acceleration graph

The instantaneous acceleration a

is the slope of the line that is tangent

to the velocity-versus-time curve at

time t

a = the slope of the velocity-versus-time graph at t

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Finding Velocity from Acceleration

Suppose we know an object’s velocity to be v1 at an initial

time t1

We also know the acceleration as a function of time

between t1 and some later time t2

We can compute the final velocity as

The integral may be interpreted graphically as the area

under the acceleration curve as between t1 and t2

𝑣2 = 𝑣1 + 𝑡1

𝑡2

𝑎 𝑑𝑡

Documents

PreClass Notes: Chapter 2 - U of T Physicsjharlow/teaching/phy131f15/cl03v...2015-06-16 2 © 2012 Pearson Education, Inc. Slide 1-3 Position and Displacement •In one dimension, position