Chapter 2Kinematics in one Dimension

April 24, 2023

Going around in circles …

A

B

s=pathlength

Seconds

x=displacement

xA

B

t=time interval (A,B) = 10 seconds

ABs

x

DEFINITIONS

s=path length or total distance traveled.

x=displacement = NET distance traveled. Difference between final and initial position independent of the path.

t=time of the trip. Usually measured in seconds.

More Definitions

average speed = distance traveled (s) / time taken [meters/sec.]

average velocity= displacement/time [m/s]

Note: the actual SYMBOL used for distance and displacement will usually change with the context of the problem.

smsm

txvelocity /5.5

1055~

10s

-40m

m/s 41040

sm

txvelocity

In the limit -

x

t txvavg

dtdx

txLimv

In the limit of t0, velocity is the slope of the tangent to the x-t curve!

vtxx

vtxx

vdtdx

vdtdxdtdxv

if

if

t

t

x

x i

f

i

0

For v=constant

Constant Velocity

v

xi

xf

x=0

xf – xi= distance traveled = v t

MOTION DOESN’T ALWAYS START AT THE ORIGIN

Question

253 2 ttx

The position of a particle as a function of time is given by theequation below.

(a) What is the position of the particle at t=2 seconds?(b) What is the velocity of the particle at t= 2 seconds?(c) At what time is the particle’s velocity = 0?(d) Is the particle ever at the origin? When?

Conclusion

Velocity is not the same at all times. It is changing A changing velocity is called an acceleration.

dtdva

tv

tvv

a initialfinalaverage

Acceleration in Pictures

Constant Velocity

Constant Acceleration

Average Acceleration

Acceleration is the rate of change of the velocity.

Dimensions are L/t2

SI units are m/s²

Example – Constant acceleration a=1 m/s2Time (s) Velocity (m/s)

0 0

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

10 10

11 11

12 12

13 13

14 14

15 15

16 16

17 17

18 18

19 19

20 20

0

5

10

15

20

25

0 5 10 15 20 25

time (s)v

(m/s

)

atvv

vvatvtva

f

initialf

0

Instantaneous Acceleration

The instantaneous acceleration is the limit of the average acceleration as t approaches 0

2

20lim x x

x t

v dv d xat dt dt

Calculus

2

2

dtxd

dtdva

dtdxv

Instantaneous Acceleration -- graph The slope of the

velocity vs. time graph is the acceleration

The green line represents the instantaneous acceleration

The blue line is the average acceleration

Acceleration and Velocity, 1

When an object’s velocity and acceleration are in the same direction, the object is speeding up

When an object’s velocity and acceleration are in the opposite direction, the object is slowing down

Example

Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same

length) Velocity is decreasing (red arrows are getting shorter) Positive velocity and negative acceleration

Some math. Assume constant acceleration

atvv

atvv

attadtaadtvvdv

adtdv

adtdv

f

tv

v

t

f

f

0

0

000 )0(

constant

0

More …

200

200

000

0

0

0

2121

0

attvxx

attvxx

tdtadtvdx

atdtdtvdx

atvdtdx

atvv

ttx

x

One more calculus trick: the the chain rulechain rule

axvv

vvxxa

vdvdxa

vdvadxdxdvv

dtdx

dxdv

dtdva

f

f

v

v

x

x

f

2

0for x

)(21)(

20

2

0

20

20

00

Kinematic Equations -- summary from the book.

Kinematic Equations

The kinematic equations may be used to solve any problem involving one-dimensional motion with a constant acceleration

You may need to use two of the equations to solve one problem

Many times there is more than one way to solve a problem

Kinematic Equations, specific

For constant a, Can determine an object’s velocity at any time t

when we know its initial velocity and its acceleration

Does not give any information about displacement

Kinematic Equations, specific

For constant acceleration,

The average velocity can be expressed as the arithmetic mean of the initial and final velocities

2xi xf

x

v vv

Kinematic Equations, specific

For constant acceleration,

Gives final position in terms of velocity and acceleration

Doesn’t tell you about final velocity

212f i xi xx x v t a t

Graphical Look at Motion – displacement – time curve The slope of the curve

is the velocity The curved line

indicates the velocity is changing Therefore, there is an

acceleration

Graphical Look at Motion – velocity – time curve The slope gives the

acceleration The straight line

indicates a constant acceleration

The zero slope indicates a constant acceleration

Graphical Look at Motion – acceleration – time curve

Let’s get real and throw something out of a building and watch it fall.This is called free fall.

Freely Falling Objects

A freely falling object is any object moving freely under the influence of gravity alone.

It does not depend upon the initial motion of the object Dropped – released from rest Thrown downward Thrown upward

Acceleration of Freely Falling Object The acceleration of an object in free fall is directed

downward, regardless of the initial motion The magnitude of free fall acceleration is g = 9.80

m/s2

g decreases with increasing altitude g varies with latitude 9.80 m/s2 is the average at the Earth’s surface

In the English/American system, g=32 ft/sec2.

Acceleration of Free Fall/Rise, cont. We will neglect air resistance Free fall motion is constantly accelerated

motion in one dimension Use the kinematic equations to solve

problems.

Free Fall ExampleLet’s drop an object from the topof this building by simply releasing it.

How long will it take to get to the ground?

How fast will it be going when it gets to theground?

What color is the object?

Issues –

Where is the origin?IS y the same thing as x?? How can that be???