Lecture 2

One-dimensional motion: displacement, velocity, acceleration

Mechanics• Kinematics deals with the concepts that

are needed to describe motion.

• Dynamics deals with the effect that forces have on motion.

• Together, kinematics and dynamics form the branch of physics known as Mechanics.

Today’s Topics:

• Introduction to Kinematics– Definitions of position, distance and

displacement– Definitions of velocity, speed, average velocity

and instantaneous velocity– Definitions of acceleration, average

acceleration and instantaneous acceleration– Graphing Motion

Position and DisplacementFirst choose a coordinate system

+ x

oo tat timeposition =x! tat timeposition =x!

ntdisplaceme =-=D oxxx !!!

SI units are meters

What’s with the little arrows?m 0.2=ox

!

m 0.7=x!

m 0.5=Dx!

m 0.5m 2.0m 7.0 +=-=-=D oxxx !!!

Distance = d = 5.0 m

m 0.2=x!

m 0.7=ox!

m 0.5-=Dx!

m 0.5m 7.0m 2.0 -=-=-=D oxxx !!!

Distance = d = 5.0 m

oxxx !!!-=D

is a vector(magnitude and direction)

d is a scalar(magnitude only)

So… direction matters!+y

+x

oxxx !!!-=D

depends only on the initial and final positions

d = distance depends on the path

x!Dx!D

ExampleA whale swims east for 6.9 km, turns around and swims west for 1.8 km and Then heads east for 3.7 km. What is the total distance travelled by the whale and its displacement?

East+ x

0=ox!

km9.61 +=Dx!

x!

You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, do you and your dog have the same displacement?

1) yes2) no

ACT: Walking the Dog

Yes, you have the same displacement. Because you and your dog had the same initial position and the same final position, then you have (by definition) the same displacement.

Follow-up: have you and your dog traveled the same distance?

ACT: Displacement

Does the displacement of an object

depend on the specific location of

the origin of the coordinate system?

1) yes

2) no

Because the displacement is

the difference between two

coordinates, the origin does

not matter.

10 20 30 40 50

30 40 50 60 70

Δx = 40m −10m = 30m

Δx = 60m − 30m = 30m

Velocity and Speed

timeElapsedntDisplaceme velocityAverage =

ttt o

o

DD

=--

=xxxv!!!

!

timeElapsedDistance speed Average =

Magnitude and Direction

( + or - )

Magnitude only

SI units are meters/sec.

Does the speedometer in a

car measure velocity or

speed?

1) velocity2) speed

3) both

4) neither

The speedometer measures speed, not velocity. Velocity is a vector (depends on direction), but the speedometer does not care what direction you are traveling. It only measures the magnitude of the velocity, which is the speed.

ACT: Speedometer

You drive for 30 minutes at 30 mi/hr and then for another 30 minutes at 50 mi/hr. What is your average speed for the whole trip?

a) more than 40 mi/hr

b) equal to 40 mi/hr

c) less than 40 mi/hr

ACT: Cruising Along I

It is 40 mi/hr in this case. Because the average speed is distance/time and you spend the same amount of time at each speed, then your average speed would indeed be 40 mi/hr.

You drive 4 miles at 30 mi/hr and then another 4 miles at 50 mi/hr. What is your average speed for the whole 8-mile trip?

a) more than 40 mi/hr

b) equal to 40 mi/hr

c) less than 40 mi/hr

It is not 40 mi/hr! Remember that the average speed is distance/time. Because it takes longer to cover 4 miles at the slower speed, you are actually moving at 30 mi/hr for a longer period of time! Therefore, your average speed is closer to 30 mi/hr than it is to 50 mi/hr.

ACT: Cruising Along II

Conceptual ProblemThe average velocity for an entire trip has a positive value. Is it possible for the average velocity over any period during the trip to have a negative value?

The average velocity for the trip is the displacement for the trip divided by the elapsed time.

It depends only on the initial and final positions, and the time required for the trip. The average velocity contains no information concerning the actual path taken by the object.

At any point in the trip, the object could temporarily reverse direction and move in the negative direction. While the object is moving in the negative direction, its velocity is negative.

As long as the overall displacement is positive, the average velocity for the trip is positive.

sm4s 2m 8 Slope +=

+=

DD

=tx

ttt o

o

DD

=--

=xxxv!!!

!

ttt o

o

DD

=--

=xxxv!!!

!

Let’s look more closely at the details

Average velocity and instantaneous velocity

t

x

ΔxΔt

tt DD

=®D

xv!

!0

lim

A

ttt o

o

DD

=--

=xxxv!!!

!

Δx

Δt

What is the velocity at point A?

Point to remember!

The velocity is related to the displacement, not the position.

So, the velocity and position can have the samesign, different signs, or even be 0.

At any given time:

AccelerationAcceleration comes into play when there is a change in velocity

ttt o

o

DD

=--

=vvva!!!

!

tt DD

=®D

vlima0

2sm6s 2

sm 12 Slope +=+

=DD

=tv

ttt o

o

DD

=--

=vvva!!!

!

Conceptual ProblemAn object moves along the +x direction with an initial velocity of +v0. Some time later it’s final velocity is +vF with | vF | < | v0 | . Is the object’s acceleration (1) positive, (2) negative or (3) 0 during this time interval?

+x

+vF+v0 | vF | < | v0 |

o

F

tt --

= 0vva!!

! is negative

Conceptual ProblemAn object moves along the -x direction with an initial velocity of -v0. Some time later it’s final velocity is -vF with | vF | < | v0 | . Is the object’s acceleration (1) positive, (2) negative or (3) 0 during this time interval?

-x

-vF-v0 | vF | < | v0 |

o

F

tt --

= 0vva!!

! is positive

Problem solving strategies1) In all calculations, write down the units explicitly.2) If you carry your units along.. You can check whether the

dimensions are correct and consistent3) Draw a figure and label it regardless of how simple the problem

may be! Most times you will have to determine relationships between variables. Visualization makes all the difference!

4) Even with all of the formulas in front of you, there is (almost) always a critical relationship or condition that you must satisfy….Think before you solve!

5) Choose a coordinate system and reference all of your kinematic variables to that coordinate system…. Keep track of signs!