x is the magnitude of a position v is the magnitude of the velocity, sometimes speed a is the...

x is the magnitude of a position

v is the magnitude of the velocity, sometimes speed

a is the magnitude of acceleration

t is time

Δ represents a change, an interval.

Δx is displacement, and can be also called “d” sometimes

Δv is a change in speed

x is the magnitude of a position

v is the magnitude of the velocity, sometimes speed

a is the magnitude of acceleration

t is time

Δ represents a change, an interval.

Δx is displacement, and can be also called “d” sometimes

Δv is a change in speed

Variables used in kinematics:

ifinitialfinal xxxxΔx ifinitialfinal xxxxΔx

ifinitialfinal vvvvΔv ifinitialfinal vvvvΔv

vΔxΔ

avx

These will be vectors, they need a direction.

These will be just numbers, or scalars.

Distance or displacement?x

This is a scalar.

It is a number, a distance, doesn’t matter the direction of

the trajectory.

This is a vector.

It is a number AND a direction.

There are special rules to operate with vectors. It is not the same as a simple number.

What’s the difference between speed and velocity?

Speed is a scalar. Scalars have only a magnitude (just a number).

For example: constant speed of a car at 60 km/h.

Velocity is a vector. Vectors have a magnitude AND a direction.

For example: velocity of an airplane traveling 200 km/h Northward.

Speed can be the magnitude of the velocity.

60 km/h v

v

v

A car goes around a curve at constant speed. Is the car’s

velocity changing?a) Yesb) No

At position A, the car has the velocity indicated by the arrow (vector) v1.At position B, the car has the velocity indicated by the arrow (vector) v2, with the same magnitude (speed) but a different direction.

Our frame of reference

(it is a necessary convention)

+

–

–

+

+3 +4

-2

-3

+4

-3-5

Quick Quizzes

Answer True or False:

1) A car must always have an acceleration in the same direction as its velocity.

2) It’s possible for a slowing car to have a positive acceleration.

3) An object with constant nonzero acceleration can never stop and remain at rest.

F

T

T

0 0 0

1 10 20

2 20 40

3 30 60

4 40 80

5 50 100

time xcar1 xcar2

__(s) (m) (m)

This is a table showing the positions of car 1 and car 2 for a motion in 5 seconds.

Which one moves further away in the same time?

Which one has the greatest average speed?

x (m)

t (s)

Displacement x time

0

Let’s plot this on the blackboard.

EQUATION OF A STRAIGHT LINE

bmxy m is the slope

b is the y-intercept (the value of y when x=0)

b

y

x

Δy

Δx

Δx

Δym

Δx

Δym

bmxy m is the slope

b is the y-intercept (the value of y when x=0)

For the equation

tx 6What is the slope?

What is the y-intercept?

For the equation

tv 317 What is the slope?

What is the y-intercept?

What does a car’s speedometer measure?

a) Average speedb) Instantaneous speedc) Average velocityd) Instantaneous velocity

A speedometer measures instantaneous speed.

Which quantity is the highway patrol more

interested in?a) Average speedb) Instantaneous speed

The speed limit indicates the maximum legal instantaneous speed.

To estimate the time a trip may take, you want to use average speed.

Passing lane

Watch the red car

Watch the blue car

POSITION versus TIME

http://www.physicsclassroom.com

In a graph POSITION versus TIME, the SLOPE indicates VELOCITY.

A steeper slope indicates greater velocity.

Passing laneWatch the red car

Watch the blue car

VELOCITY versus TIME

In a graph VELOCITY versus TIME, constant velocity is a straight line.


In general,

Average ≠ Instantaneous

Is the instantaneous velocity at point A greater or less

than that at point B?

a) Greater thanb) Less thanc) The same asd) Unable to tell from this graph

The instantaneous velocities can be compared by looking at their slopes. The steeper slope indicates the greater instantaneous velocity, so the velocity at A is greater.

In the graph shown, during which time interval is the

acceleration greatest?a) Between 0 s and 2 s.b) Between 2 s and 4 s.c) Between 4 s and 8 s.d) The acceleration does not change.

The acceleration is greatest between 2 s and 4 s. The velocity is changing fastest, and the graph has the greatest slope, during this interval.

At which point is the magnitude of the

acceleration the greatest?a) Point Ab) Point B c) Point C d) The acceleration does not

change.

The magnitude of the acceleration is greatest when the velocity is changing the fastest (has the greatest slope). This occurs at point A.

2.5 CONSTANT ACCELERATION Important equations!

Many applications of mechanics involve objects moving with constant acceleration.

Constant “a ” means that

average ā = instantaneous a

For convenience, let’s adopt vf = v and vi = v0 , ti = 0 , tf = t so:

t

vv a 0

t

vv a 0

atvv 0 atvv 0or

Because velocity is increasing or decreasing UNIFORMLY with time, we can express:

20 vv

v

2

0 vvv

averagefinal

Remember that: or Δt

Δxv

Δt

Δxv ΔtvΔx ΔtvΔx 3

1

2

So now I will substitute inside , and then again inside the resulting equation.

2

3 1

tavv if tavv if

Graph will be a straight line.

Graph will be a straight line if a = 0,

but it will be a parabola if a ≠ 0.

Velocity depends on t

Position depends on t and t2

Graph v vs t Graph x vs t

From a graph: slope or area?

Example:

Slope

Area

Example:

What should you do if you have a velocity versus time graph and you want….

acceleration ?

displacement ?

AREAS

v (m/s)

t (s)

SLOPE

During which time interval is the distance traveled by the

car the greatest?a) Between 0 s and 2 s. b) Between 2 s and 4 s. c) Between 4 s and 6 s. d) It is the same for all time

intervals.

The distance traveled is greatest when the area under the velocity curve is greatest. This occurs between 2 s and 4 s, when the velocity is constant and a maximum.

Important: our equations can only be used in regions of CONSTANT

ACCELERATION.

Constant velocity in the positive direction

20 21 taΔtvΔx 20 21 taΔtvΔx tavv 0 tavv 0 tva tva

any tat 0a any tat 0atm/s) (12x tm/s) (12x 12m/svv 0 12m/svv 0



any tat 0a any tat 0atm/s) 12(x tm/s) 12(x 12m/svv 0 12m/svv 0


Constant velocity in the negative direction

CONSTANT POSITIVE ACCELERATION


2m/s 1,7a 2m/s 1,7a2)27.1(x t 2)27.1(x t t)m/s 1,7(v 2 t)m/s 1,7(v 2


CONSTANT NEGATIVE ACCELERATION


2m/s 1.7a 2m/s 1.7a2)27.1(20x tt 2)27.1(20x tt t 1.720v t 1.720v


t

v0

t0

Vel

ocity

v

Acc

eler

atio

n

0

a

t

v(t)

a(t) = 0

Slope = 0

Pos

ition

0

x0

x

x(t)

Slope = v0

Pos

ition

0

x

t

v0

t0

Vel

ocity

v

Acc

eler

atio

n

0

a

a0

t

x(t)

a(t)

v(t)

Slope = 0

Slope = a0

x0

Slope varies

200 2

1attvxx(t) 2

00 2

1attvxx(t) tvxx(t) 00 tvxx(t) 00

atvv(t) 0atvv(t) 0

0vv(t) 0vv(t)

0aa(t) 0aa(t) 0a(t) 0a(t)

UNIFORM MOTION (velocity is constant)

UNIFORM ACCELERATION(acceleration is constant)

Quick QuizParts (a), (b), and (c) of the figure below represent three graphs of the velocities of different objects moving in straight-line paths as functions of time. The possible accelerations of each object as functions of time are shown in parts (d), (e), and (f). Match each velocity vs. time graph with the acceleration vs. time graph that best describes the motion.

1. a and e, b and f, c and d

2. a and d, b and f, c and e

3. a and e, b and d, c and f

0 2 4 6 8 10Time (s)

0

1

2

3

4

5

Dis

tanc

e (m

)Match this plot:

0 2 4 6 8 10Time (s)

0

1

2

3

4

5

Vel

ocity

(m

/s)

(a)

0 2 4 6 8 10Time (s)

0

1

2

3

4

5

Acc

eler

atio

n (m

/s2)

(b)

0 2 4 6 8 10Time (s)

-2

-1

0

2

Vel

ocity

(m

/s)

(c)

0 2 4 6 8 10Time (s)

0

1

2

3

4

5

Vel

ocity

(m

/s)

(d)

0 2 4 6 8 10Time (s)

0

1

2

3

4

5

Acc

eler

atio

n (m

/s2 )

(e)

1

3

0 2 4 6 8 10Time (s)

0

1

2

Acc

eler

atio

n (m

/s2)

Match this plot:

2 4 6 8 10Time (s)

0

4

Vel

ocity

(m

/s)

(b)

0 2 4 6 8 10Time (s)

0

1

2

3

4

5

Vel

ocity

(m

/s)

(d)

-1

-2 0 2 4 6 8 10Time (s)

Dis

tanc

e (m

)

(a)

10

0

20

0 2 4 6 8 10Time (s)

15

Dis

tanc

e (m

)

(c)

10

5

0

-5

20

.

0 2 4 6 8 10Time (s)

25

Dis

tanc

e (m

)

(e)

20

15

10

5

30

8

6

2

0

-2

30

40

50

60

Acceleration of gravity does not depend on the mass

VACUUM AIR

Gravitational acceleration g does NOT depend on the weight of the object.

Apollo 15 Moon walk in 1971.

Commander David Scott

Equations governing falling objects and objects thrown upward

200 2

1gttvyy(t) y 2

00 2

1gttvyy(t) y

gtv(t)v yy 0 gtv(t)v yy 0

What is the ball’s acceleration at

the top of its path (at t=2 s)?

a) zero.b) +9.8 m/sc) -9.8 m/sd) +9.8 m/s2

e) -9.8 m/s2

Gravity does not “turn off” at the top! The ball’s velocity is still changing, as it changes from going up to going down. For a moment the velocity is zero, but the gravitational acceleration is a constant throughout the path.

t = 0

y = 0

v = + 20 m/s

t = 1 s

y = 15 m

v = + 10 m/s

t = 2 s

y = 20 m

v = 0

t = 3 s

y = 15 m

v = –10 m/s

t = 4 s

y = 0 m

v = –20 m/s

a = -g

v

v

v

va

a

a

a

a

Always:

Let’s suppose now that the initial velocity of the ball upward is +35 m/s (instead of the +20 m/s as in the picture).

What is the maximum height that the ball will reach?

Assume that g = 10 m/s2

Maximum height ymax = 20 m

Equations we can use:

gtvv 0 gtvv 0

200 2

1gttvyy 2

00 2

1gttvyy

Answer: ymax = 61 m

Quick Quizzes

A tennis player on serve tosses a ball straight up. While the ball is in free fall, its acceleration

1. increases. 2. decreases. 3. increases and then decreases.

4. decreases and then increases. 5. remains constant.

A tennis player on serve tosses a ball straight up. As the tennis ball travels through the air, its speed

1. increases. 2. decreases. 3. increases and then decreases.

4. decreases and then increases. 5. remains constant.

x is the magnitude of a position v is the magnitude of the velocity, sometimes speed a is the...

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