Kinematics in One Dimension -...

Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI

Lecture 2

Chapter 1 and 2

Kinematics in One Dimension

Physics I

ResponseWare App instead of a clicker


Today we are going to discuss basic kinematic stuff:

Distance/Displacement: Section 1.3

Speed/Average velocity: Section 1.4 Instantaneous velocity: Section 2.2

Average Acceleration: Section 1.5 Instantaneous Acceleration: Section 2.7


Mechanics

There are three branches of Mechanics:Kinematics Motion Forces Statics Motion ForcesDynamics Motion Forces

Kinematics describes motion of objects we are not interested in reasons (forces) of a motion

Motion


Kinematics


Frames of Reference

-5 5

-5

5 y-axis

x-axis

Before starting solving any problem, we have to define a coordinate system (a frame of reference) to describe position and motion of an object

In this class, we will base problems in a Cartesian coordinate system. We will have 1D and 2D problems

We have freedom to choose direction of an axis. For 1 dimensional (1D) motion (motion in a straight line), it’s better to align the x-axis along a

motion direction. For falling bodies, we tend to describe position using the y-axis

origin

1 2 3 4 5

Positive direction+x

Negative direction-x

x

One dimensional (1D) problem Two dimensional (2D) problem

origin


Distance vs. Displacement


Distance vs. Displacement

Distance (scalar): the total path length traveled by an object

Displacement (vector): how far an object is from its starting point

What would be your displacement after a complete roller coaster?

ConcepTest 1 Roller Coaster

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?

A) yes

B) no

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.

ConcepTest 2 Walking the Dog

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

Follow-up: how would you measure displacement in your car?

Does the odometer in a car

measure distance or

displacement?

A) distance

B) displacement

C) both

ConcepTest 3 Odometer

If you go on a long trip and then return home, your odometer does not measure zero, but it records the total miles that you traveled. That means the odometer records distance.


Distance vs. Displacement (1D)Distance is a scalar

Displacement is a vector– A vector has both magnitude and direction (or sign in 1-D)

Displacement =x2- x1=+40 m

Distance =

20 40 60 70

x2

x

x1

70+30 =100 m

Displacement = final position – initial position


Distance vs. Displacement (1D)

x1 x2

Distance = 20 m

x1

Distance = 20 m

Displacement =negative

Displacement =

x1

30-10= + 20 m

x2

10-30= -20 m


Speed and Velocity

Even Hollywood feels that there is a difference between these two terms


Average Speed and Average Velocity

elapsed time travelleddistance

speedaverage

(Speed: Distance traveled per unit time interval)

Speed is a scalar

Velocity is a vector

(Velocity: Displacement of an object per unit time interval)

elapsed timentdisplaceme

velocityaverage

Does the speedometer in a car

measure velocity or speed?

A) velocity

B) speed

C) both

D) neither

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

ConcepTest 4 Speedometer


Graphs: Average velocity

0 1 2 3 4 5 60

5

10

15

20po

sitio

n (m

)

time (s)

∆t

∆x

t1 t2

x1

x2


Instantaneous VelocityAverage velocity does not tell the whole story…we also need

If you watch a car’s speedometer, at any instant of time, the speedometer tells you how fast the car is going at that instant.


Instantaneous velocity

0 1 2 3 4 5 60

5

10

15

20po

sitio

n (m

)

time (s)

t1

goes to 0




Graphically, instantaneous velocity is the slope of the x vs t plot at a single point

Mathematically, the instantaneous velocity is the derivative of the position function


Finding instantaneous velocity from position graphically

.

>0

Turning point

Moves forward Moves backward

0

v positive

It flies back.v4 negative

v1 (slow speed)v2 (max speed)

v3=0 (turning point)

Steeper slope ≡ higher velocityGentler slope ≡ lower velocity

Which one of the following x-vs-t graphs could be a

reasonable representation of the motion of a baton in

a relay race being passed from one runner to the next?

ConcepTest 5 Relay baton

End of class


Finding Position from a Velocity Graph

The total displacement ∆s is called the “area under the velocity curve.” (the total area enclosed between the t-axis and the velocity curve).

The displacement is the shaded area

∆

Example

∙ Let’s integrate it: ∙ x ∙

∙

Initial position

Geometrical meaning of an integral is an area

Total displacement

displacement

v(t)

t

Here is the velocity graph of an object that is at the origin (x = 0 m) at t = 0 s.At t = 4.0 s, the object’s position is

A) 20 m

B) 16 m

C) 12 m

D) 8 m

E) 4 m

ConcepTest Position from velocity

Displacement = area under the curve


Acceleration


AccelerationVelocity can also change with time: acceleration

elapsed time velocityof change

onacceleratiaverage

Speeding up: acceleration

Instantaneous acceleration

If we are given x(t), we can find both velocity v(t) and acceleration a(t) as a function of time

Slowing down: deceleration


A particle is moving in a straight line so that its position is given by the relation

x = (2 m/s2)t2 + (3 m). Calculate

(a) its average acceleration during the timeinterval from t1 = 1 s to t2 = 2 s,

(b) its instantaneous acceleration as afunction of time.

Example 2-7: Acceleration given x(t).


Thank youSee you on Wednesday

Kinematics in One Dimension -...

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