Chapter 2Kinematics – Rectilinear Motion

•Displacement, Velocity & Acceleration

•Equations describing motion along straight line

•Objects falling freely under earth‟s gravity

MECHANICS

DYNAMICS(Force & Response)

STATICS(At rest)

KINEMATICS(describes motion)

Concepts to study objects in motion

Position

Displacement (similar to distance)

Velocity (similar to speed)

Acceleration

What is “Position” in Physics?

Frame of reference

A choice of coordinate axes with an origin

Coordinates for representing starting and ending points of a motion

One dimension only (this chapter)

(example) a straight line extending to the right and left of the point at the center (origin) and called the x direction

Displacement

Defined as the change in position

f - final position; i - initial

Units -meters (SI)

CAN BE POSITIVE OR NEGATIVE ALONG A GIVEN DIRECTION !!!

f ix x x

From A to B xi = 30 m xf = 52 m x = 22 m The displacement is

positive, indicating the motion was in the positive x direction

From C to F xi = 38 m xf = -53 m x = -91 m The displacement is

negative, indicating the motion was in the negative x direction

Some examples of displacement

Initial x Final x ∆xSign of ∆x

30 52 22 +

38 -53 -91 -

-50 -30 20 +

42 31 -11 -

Displacement is a “vector”

(Need both magnitude (size) and direction to completely describe)

For 1 D motion just + or – sign

(Scalar quantities have just magnitude)

IMPORTANT NOTE

Motion along one dimension may be either along x or y or z axis direction and can be used to represent horizontal side to side (motion along x axis) or front and back (motion along z axis) or vertical up and down motion (motion along y axis)

Displacement not always ≠ distance

Throw a ball straight up and then catch it at the same point you released it

Distance = 2 times height reached by ball

The displacement is zero!!!!!!

so is zeroi fx x x

Two dimensions Example

East

Walk 3 blocks east and then 4 blocks north

Total distance = ?

Displacement = ?

Speed

The average speed of an object is defined as the total distance traveled divided by the total time elapsed

SI units – meters/second m/s

Speed - scalar? Vector?

Speed always positive?

total distanceAverage speed

total time

dv

t

You drove to Trenton campus 8 miles. You drove for 17 minutes and stopped for 3 mins. at a gas station on your way. What was your :

Displacement?

Distance covered?

Average speed?

+8 miles relative to origin at WWC

8 miles

0.4 miles/min or 10.7 m/s

Think!

What is the displacement for the round trip if you returned back to college?

Velocity

Average velocity =

SI units – m/s (other units feet/s, cm/s)

Velocity can be positive or negative

Velocity is a vector! For 1D, just +ve or negative

Direction of velocity same as displacement

f iaverage

f i

x xxv

t t t

displacement

time elapsed

You drove to Trenton campus 8 miles. You drove for 17 minutes and stopped for 3 mins. at a gas station on your way. What was your average velocity?

Average velocity =+ 10.7 m/s

What is the average velocity for the round trip?

Speed vs. Velocity

average velocity = average speed?

Note: In text, bars over physical quantities represent average values.

e. g. average velocity, average speed, average acceleration

Quick Quiz 2.1

Total distance =

Displacement =

Average velocity in x direction =

Average Speed =

200 yards, 0, 0, 200÷25 = 8 yards/sec

Chapter 2 Lecture 2

understand changing velocity & acceleration;

motion with constant acceleration

special case of constant acceleration: free fall

Understanding Velocity & Acceleration from Graphs

Position(x) vs. time (t) & velocity(v) vs. time (t) graph

Object moving with a constant velocity (uniform motion) – what is the shape of x –t graph?

Average velocity = slope of line joining initial & final positions

Slope of a line is rise divided by run

change in vertical axisslope

change in horizontal axis

comparing constant & changing velocity

equal displacements in equal intervals of time; average velocity is constant all the time; uniform motion

average velocity (slope of curve) changes from time to time; change in velocity means acceleration!

Instantaneous velocity

For non uniform velocity, use instantaneous velocity (called simply v) instead of vavg

The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero

instantaneous velocity indicates what is happening at every point of time when velocity keeps changing

lim

0t

xv

t

Instantaneous Velocity on a Graph

The slope of the line tangent to the position-vs.-time graph is defined to be the instantaneous velocity at that time

The instantaneous speed is defined as the magnitude of the instantaneous velocity

Acceleration

This is average acceleration

Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Customary)

Vector? Yes, in 1D + or -

f i

f i

v vva

t t t

change in velocityAcceleration

time elapsed

What are vi and vf ?

Instantaneous values of velocities (initial and final)

Class Example 1

A car takes 2 seconds to accelerate from an initial velocity of +10 m/s to a final velocity +20 m/s. What is the average acceleration?

ā =+5 m/s2

acceleration is positive and in the same direction as initial velocity

Class Example 2

A plane travelling at +30 m/s on the runway stops in 5 seconds. What is the acceleration?

ā = -6 m/s2

(a is negative and it has a sign opposite to that of initial velocity)

TRUE OR FALSE

“So, to slow down an object you always

need to have negative acceleration. To speed up an object, you always need to have positive acceleration.”

FALSE, FALSE, FALSE, FALSE, FALSE, FALSE

Example: Motion Diagram of A Car

Please insert active figure 2.12

Average Acceleration

When the sign of the initial velocity and the acceleration are the same (either positive or negative), then the speed is increasing

When the sign of the initial velocity and the acceleration are in the opposite directions, the speed is decreasing

Negative Acceleration

A negative acceleration does not necessarily mean the object is slowing down

If the acceleration and velocity are both negative, the object is speeding up

Finding Average Acceleration From v-t graph

Relationship Between Acceleration and Velocity

Uniform velocity (shown by red arrows maintaining the same size)

Acceleration equals zero

Relationship Between Velocity and Acceleration

Velocity and acceleration are in the same direction

Acceleration is uniform (blue arrows maintain the same length)

Velocity is increasing (red arrows are getting longer)

Positive velocity and positive acceleration

Relationship Between Velocity and Acceleration

Acceleration and velocity are in opposite directions

Acceleration is uniform (blue arrows maintain the same length)

Velocity is decreasing (red arrows are getting shorter)

Velocity is positive and acceleration is negative

Quick Quiz 2.2

True or False?

a) A car must always have an acceleration in the same direction as velocity

b) It is possible for a slowing car to have +veacceleration

c) An object with constant nonzero acceleration can stop but then can not remain at rest

Answers:

a. Think about class example 2

b. Think about slowing car with initial velocity of -10 m/s and final velocity -5 m/s

c. Think about +ve velocity, -ve acceleration of chalk thrown vertically upward

1D motion with const. acceleration

Equations with constant acceleration, a

Objects thrown up or down on earth move with a special constant acceleration called “g” the acceleration due to gravity = 9.8 m/s2

2

2 2

1

2

1

2

2

o

o

o

o

v v at

x vt v v t

x v t at

v v a x

Notes on the equations

t2

vvtvx fo

average

If initial t=0 & final t=t, then ∆t = t

Gives displacement as a function of velocity and time

Use when you don‟t know and aren‟t asked for the acceleration

Notes on the equations

Shows velocity as a function of acceleration and time

Use when you don‟t know and aren‟t asked to find the displacement

ov v at

Notes on the equations

Gives displacement as a function of time, velocity and acceleration

Use when you don‟t know and aren‟t asked to find the final velocity

2

o at2

1tvx

Notes on the equations

Gives velocity as a function of acceleration and displacement

Use when you don‟t know and aren‟t asked for the time

2 2 2ov v a x

Textbook Example 2.4, page 37, Daytona 500

Starts from rest: at t=0, v is zero (v0 = 0)

Constant acceleration: uniform a = 5m/s2

Velocity of car after a distance of 100 ft =?

Use equation:

For distance convert feet to m! (1 m = 3.281 feet)

Time elapsed=?

Use equation:

2 2

02a xv v

0v atv

Textbook example 2.4 Ans: 17.5 m/s & 3.50 s

End of Chapter MC1:

(!!! sig fig & units!!!)

Choose upward vertical direction as +ve y axis

(coming up: free fall, use „y‟ axis)

Initial velocity +15

Final velocity -8

(Assume a = -9.8 ….coming up)

Find t

Use 0

v atv

2.35 s

Problem 9

Instantaneous velocities of tennis player at

a) 0.50 s

b) 2.0 s

c) 3.0 s

d) 4.5 s

Just find slope of graph around each t value

4, -4, 0, 2 with sig fig & units

Galileo Galilei

1564 - 1642

Galileo formulated the laws that govern the motion of objects in free fall

Also looked at:

Inclined planes

Relative motion

Thermometers

Pendulum

Free Fall

All objects moving under the influence of gravity only are said to be in free fall

Free fall does not depend on the object‟s original motion

All objects falling near the earth‟s surface fall with a constant acceleration

The acceleration is called the acceleration due to gravity, and indicated by g

Acceleration due to Gravity

Symbolized by g

g = 9.80 m/s² pointing towards the earth g 10 m/s2

g is always directed downward Toward the center of the earth

Ignoring air resistance and assuming gdoesn‟t vary with altitude over short vertical distances, free fall is constantly accelerated motion

Who is in free fall?

Space shuttle!!!

Free Fall – an object dropped

Initial velocity is zero

Let up be positive

Use the kinematic equationsTip: Generally use y

instead of x since vertical

Acceleration is g= -9.80 m/s2

vo= 0

a = g

Free Fall – an object thrown downward

a = g = -9.80 m/s2

Initial velocity 0

With upward being positive, initial velocity will be negative

Free Fall -- object thrown upward

Initial velocity is upward, so positive

The instantaneous velocity at the maximum height is zero

a = g = -9.80 m/s2

everywhere in the motion

v = 0

Thrown upward, cont.

The motion may be symmetrical

Then tup = tdown

Then v = -vo

The motion may not be symmetrical

Break the motion into various parts

Generally up and down

Problem 45: thrown upward with speed 25

This is free fall motion with uniform a=g=9.8

Upward direction is +ve y axis & use g=-9.8

a) Maximum height reached

At this value of ∆x, final v = 0; initial v = 25

Use

b) Use

2 2

02a yv v

0v atv

31.9m, 2.55s, same t, same initial v & opposite

Class Example

Obama throws a rock down with speed of 12 m/s

from the top of a tower. The rock hits the ground after 2.0s. What is the height of the tower?

Choose down as positive & choose ∆y for displacement

Given initial velocity (v0=12)

Given time (t = 2)

Given free fall (moving under constant downward acceleration (a=g=9.8)

Asked to find ∆y - the distance traveled by rock from top to bottom (height of tower)

Use 2

0

1

2y t av t

44 m (rounded)

In free fall motion of chalk, there is constant acceleration called „g‟ of magnitude 9.8 meter per second squared throughout its trip from the time it leaves my hands until it hits the ground

Non-symmetrical Free Fall

Need to divide the motion into segments

Possibilities include

Upward and downward portions

The symmetrical portion back to the release point and then the non-symmetrical portion