What you should know from Chap. 1:

Def. of physics

major subfields

Scientific Method

Concept of Measurement and uncertainty

Significant figures

Standards of length, mass, and time

SI, cgs, british units

Powers of 10 (including prefixes)

Vectors (definition, graphic analysis, components, unit vectors)

Dimensional Analysis

What is: 10-12 boo

109 low

10-6 scope

1018 boy

106 phone

10-9 goat

1012 bull

Linear Motion

Heros: Galileo (1564-1642) and Newton (1642-1727).

Mechanics - the study of motion, forces, and energy; their relations.

Kinematics -- description of motion.

Dynamics -- relations of forces and energy to produce motion.

To describe motion need a coordinate system

1. Fixed reference point

2. Axes with scales

3. Labels on axes

Vectors and Scalars

Scalar: a quantity that can be described by a single number.

24 apples in a box

30 degrees F outside

5 ‘ 9 ‘‘ tall (1.75 meters)

                        110 I.Q.

Vector: a quantity described by a magnitude and a direction.

airplane heading 520 mph due west

mother lives 2400 miles due east from Pullman

rocket went straight up 3 miles from Mercer Is.

Described by arrows with arrow pointing in the direction of the vector and the length representing the magnitude.

DISPLACEMENT: a vector describing the net change in position.

One Dimensional (Linear, Rectilinear) motion takes place in a straight line:

magnitude of displacement = distance

direction is "to the right" or along the +x direction.

Average Velocity

For a given displacement, Dx, occurring in a time interval Dt, the average velocity, vavg = Dx/Dt. UNITS: L/T {m/s, km/hr, miles/hr}

(Must state direction of vavg which = direction of Dx).

 

Another way to write vavg : Let 1 and 2 represent "initial and final" -- then,

t2 - t1 > 0, but x2 - x1 can be both ± ( or 0).

Therefore vavg can be ± (or 0).

Speed

The magnitude of vavg = average speed. UNITS: L/T.

speed = distance traveled/time elapsed

Speed is only + (or 0).

At Indianapolis Speedway, cars average 200 mph. Interpret in terms of speed and average velocity.

speed = 200 mph; average velocity = 0.

Some Speeds

 

light 3.0 x 108 m/s

recession of fastest known quasar 2.8 x 108 m/s

electron around nucleus (H atom) 2.2 x 106 m/s

Earth around Sun 3 x 104 m/s

Rifle bullet (muzzle velocity) ~7 x 102 m/s

Random motion of molecules in air 4.5 x 102 m/s

Sound (in air) 3.3 x 102 m/s

Jet plane (Boeing 747) 2.7 x 102 m/s

Cheetah (all out) 28 m/s

Human (all out) 12 m/s

Human walking (with vigor) 1.3 m/s

Snail (all out) ~10-3 m/s

Glacier 10-6 m/s

Rate of growth of hair (human) 3 x10-9 m/s

Continental Drift ~10-9 m/s

Instantaneous Velocity

Often plot v vs. time:

As Dt ---> 0 , Dx/Dt approaches the slope to the x(t) curve. This limit is the instantaneous velocity at the single point where the tangent to the curve touches the curve.

vinstantaneous = v = lim Dx/Dt = dx/dt

Dt-->0

Plot of position vs time for displacement in + x direction for three types of motions:

In practice, to determine v need either:

(a) a detailed graph of x(t)

(b) a detailed table of values of x vs. t

(c) an equation for x(t) -- then we use numerical methods to calculate v or use "derivative" of x with respect to time.

In language of calculus -- velocity is the rate of change of displacement with respect to time. (speed is the rate of change of distance with respect to time): dx(t)/dt.

Estimate distance to get to class _______ miles; Time_____ min. Find speed in miles/min, miles per hour, m/s

speed = distance/time

 

 

1 mile/minute = 60 miles/hr

 

 

1 mile/minute x (1609 meters/mile x 1 min/60 sec) = 26.8 m/s

 

 

 

Car has speedometer, odometer, and a clock. Your very bored passenger is fanatic about numbers and records all three on a trip from Pullman to Seattle. "Display" the data and discuss.

Display means to plot on reasonable axes -- distance vs time and speed vs time.

A milk train leaves Seattle at 4 am and travels at a constant speed of 30 mph. An express train leaves Seattle on a parallel track at 5 am and travels at a constant speed of 90 mph. How long will it take the express train to overtake the milk train.

[asks for TIME].

Conceptualize motion with a graph:

We know that ; so what is d?

Same distance traveled by milktrain d = dexpress = dmilk

BUT dmilk = vmilk . tmilk where tmilk = Dt(= 1 hr) + texpress

d = vmilk ( Dt + texpress) or

 

 

where Dt= 1 hr, vmilk = 30 mph; vexpress = 90 mph, so texpress = 0.5 hr;

Therefore the express train overtakes the milk train in 1/2 hr.

The Two Bikers and the Bee

Two bikers travel at uniform speeds of 10 mph towards each other. At instant they are 20 miles apart, a bee flies from the front wheel of one of the bikes at a uniform speed of 25 mph directly to the front wheel of the other bike. It touches it and turns around (instantly) and returns at the same speed to the front wheel of the first bike, touches it, turns around (instantly) --- back and forth -- until the front wheels collide and make bee jelly!

What was the total mileage covered by the bee in its back and forth journey?

Intuitively, the time from 20 miles separation to collision is 10 miles/10 mph = 1 hour.*

*

*

*

****

**

*

(the bikes collide in the center)

Since the bee takes no time to turn around, it travels 25 mph x 1 hour = 25 miles.

The non-intuitive approach: Biker1 position = vt

Biker2 position = 20 - vt

At t = tcollision, Biker1 position = Biker2 position, or

vtcollision = 20 - vtcollision

Very non-intuitive method: Try to add up the entire back and forth motion. NO WAY!!

Two Bikers and the Bee "Plot"

What you should have been doing:

Reviewing class notes and filling in

Reading Sec. 2-9 -- suggests How To Solve Problems

Answering Chap. 2 Questions

Working problems in Chap 2

Carefully going over examples in Chap. 2

Reading and thinking about acceleration

What you should know and understand:

Conversion factors

scalar vs vector quantities

definitions of:

displacement

distance

avg velocity

avg speed

instantaneous velocity

instantaneous speed

THE DIFFERENCE BETWEEN:

SPEED AND VELOCITY

DISTANCE AND DISPLACEMENT.

Which are scalars, which are vectors:

volume of a cube s

wind speed s

wind speed and direction v

GPA s

water flow in a river v

location of significant quantities of cookies rel. to you v

tire pressure s

Acceleration

Any motion with a change of velocity is accelerated motion.

Car speeding up from rest -- accelerating.

Car breaking from a constant speed down to 0 -- accelerating (often say de-accelerating to mean acceleration is < 0).

VELOCITY CHANGES. Let v1, t1 v2,t2 be velocity and time before and after a velocity change.

average acceleration = change of velocity/time interval

t2 > t1 so t2 - t1 > 0;

ACCELERATION IS A VECTOR!!!

For one dimensional motion often use sign of Dv to indicate direction of acceleration.

instantaneous acceleration = limit of aavg as Dt --> 0:

ainstantaneous = a = lim Dv/Dt

Dt-->0

Here Dv is an infinitesimally small change in v during the infinitesimally small change in time, Dt. (rate of change of velocity with respect to time "C" word = derivative of v wrt time).

IS ACCELERATION THE SAME THING AS VELOCITY????

NO!!!!

Do they have different units? YES:

Are they usually unequal in magnitude and direction? YES.

THEREFORE THEY CANNOT BE THE SAME THING.

What if speed is constant but direction of v changes?

Change of direction only: still ----> ACCELERATION.

Indianapolis Speedway example: car is accelerating because direction of v changes (continuously):

 

What you should have been doing:

Reviewing class notes

Working problems galore.

What you should know and understand:

definitions of:

average acceleration

instantaneous acceleration

initial conditions (xo, vo) --

simple: values of position x and velocity v (both magnitude and direction) at t=0.

THE DIFFERENCE BETWEEN:

velocity and acceleration

how to interpret a graph of x vs t, v vs t

two simple examples

 

 

 

hobbes cartoon

 

 

 

A sled with two riders accelerates down a uniformly slope from rest to 100 m/s in 10 s. What is the acceleration?

v1 = 0, t1 = 0, v2 = 100 m/s, t2 = 10 s ---->

down the hill.

IF a = aavg, then the velocity would smoothly increase 10 m/s every second for 10 seconds, reaching 100 m/s.

A racing car traveling along +x axis deploys a parachute at some instant while traveling 200 km/hr. The car experiences a constant de-acceleration of 25 km/hr/s for 5 seconds. What is the car’s velocity at 5 seconds.

Since a = constant, can use

t1 = 0; t2 = 5 s; v1 = 200 km/hr; aavg = -25 km/hr (i.e., pointing in -x direction). So:

along +x direction.

Uniformly Accelerated Motion (a = constant)

Let t1 = 0 and t2 = t, the elapsed time.

Let x1 = xo (initial position at t=0) and x2 = x (position at t)

Let v1 = vo (velocity at t=0) and v2 = v (velocity at t)

Then:

FOR a (constant) = aavg

Therefore: (for constant a)

For a constant, (ONLY FOR a constant).

Eliminate v:

{constant a}

warning: in solving for t: usually two solutions; t cannot be imaginary.

Eliminate t:

{a constant}

warning: solving for v: usually two solutions; v cannot be imaginary.

For {a constant} can eliminate a (see above) :

(x - xo)

SONAR (POSITION SENSOR)

 

 

FREE FALL and Related Problems

{good approx. for very dense objects (stones, balls, bullets)}

Time to fall 2 m: Let xo = 0, vo = 0; use +g

Arrow Shot Straight Up

Powerful bow -- launches arrow straight up at 90 m/s.

• How high will the arrow rise?

KNOW: vo (= 90 m/s), xo (0 m), a (= -g - 9.81 m/s2)

Don’t KNOW: x-xo, t

Predict shape of v(t) and x(t):

At what instant (value of t) does it reach the top:

• At what instant (value of t) will it return to the ground?

exactly 2 x time to reach the top!! (motion up very symmetric w.r.t. motion down).

• What is the velocity of the arrow when it hits the ground?

GUESS.

• WHAT IS THE ACCELERATION AT THE TOP?

v = 0 (so what?)

a = -g!!! constant -- means always!

Falling Flower Pot

A flower pot from a ledge above falls past your window while you happen to be watching with a timer. You determine that it takes 0.2 s for the pot to fly past your window, which is 4 m high. How far above the top of the window is the ledge from which the pot fell?

Falling Meter Stick

Given g, use a meter stick to measure reaction time tr.

 

 

Dollar Bill

A High School Physics Teacher wrote me (jtd) the following email message regarding a ball thrown straight up; what is the velocity and acceleration at the top?

"Concerning the zero velocity at the ball's peak, I have my students plot a series of displacement vs time, velocity vs time, then acceleration vs time graphs with data for an object shot into the air. A rather tedious activity, but they do come away convinced that at the peak, v=0 and also that even at that instant, a ˜ -10 m/s2."

v initially is up; after it turns around v is down; at some instant v = 0!

Take v up as + , we can plot v vs. t; cannot switch directions without going though 0:

"It's hard for them to buy that an object could have a ? 0 when the velocity is 0, revealing to me their limited grasp of acceleration."

[NOT MY STUDENTS!!!! ---- T.D.]

acceleration is a constant for all times during flight of ball; a = -g

"They basically have a hard time turning loose of the notion with which they come into physics: to accelerate is to speed up only."

any change in velocity means a ? 0; incl. ± and direction.

So think about this one:

You throw a ball straight towards an oncoming locomotive and the ball bounces elastically. Describe qualitatively what happens. Does the ball ever experience v = 0?

In frame of reference of earth, vball will go to zero; slightly later in time vball in locomotive frame also = 0; either case, v passes through 0.