PART 1: INTRODUCTION

Mechanics is the branch of Physics that deals with the motion of objects and the forces that change it. Mechanics is divided into 2 areas called kinematics and dynamics .
Kinematics describes the motion of objects without reference to the forces that act on them. Dynamics, on the other hand, is the study of the explicit relationship between forces and their effect on motion.
Galilei Galileo is the one who introduced this branch of Physics. He did study Free Fall
and motions along incline planes. He described the motion of objects in free-Fall.
(see the bug and the elephant. we neglect the presence of air).


PART2: DISPLACEMENT

The displacement vector ds is a vector that points from the initial position to the final position.
A displacement is a change in position.
1)  Take the number line. Imagine a bug crawling on the number line.What is the change in position from A to B ? (what is the displacement of the bug from A to B)
i)

ds = ________   (a negative sign = @ left)

hint: you can trace a vector from A to B and see that ds = 6 steps@left = -6
or you can find the change in position xB = -2   xA = 4 so xB - xA = -2 -4 = -6
ii)

ds = ________   (a negative sign = @ left)
iii)

ds = ________   (a negative sign = @ left

iv)

ds = ________   (a negative sign = @ left

2) the bug moves from A to D, but makes a stop along the way at B, C

ds = ________   (a negative sign = @ left

(hint: you can trace a vector (arrow) from A to D (initial to final) and count the magnitude of the
vector and see its direction. Or you can do ds = xD - xA  (final - initial) )

3) same question as 2)

4) same as 2)

If you want to find the displacement of the bug between its starting point A at its finish point B, just trace an arrow from A to B. The path taken by the bug does not matter.
If you compute the distance, the path does matter.


We will only deal with motion along a straight line. In such a case, a displacement that points to the West is assigned a positive value, and a displacement in the opposite direction is assigned a negative value.

5) A fish swims 500m to the West . ds =
A fish swims 500m to the East. ds =



PART III: AVERAGE SPEED

The average speed of an object is defined as the distance traveled divided by the amount required by the amount of time required to cover this distance. Vav = ds / dt  .   ds is the distance.
0) Here is a short review on conversion. You have 2 ways to do it:
Say a speed is 443 m/s and you want to convert to km/h
Here is 1 way to do that
A) Follow the step : 443 m = 0.443 km
                            1 s =  0.000278 hours  (1 hour is 3600 s so 1 s = 1/3600  s )
so 443 m/s =0.443 km /0.000278 hours = 1594 km/h (about)

B) 1km/1000m = 1   or 1000m/1km= 1    and 1hr/3600s =1      or       3600s/1hr =1
first let's convert m to km        443m /s   =  443 m/s   x 1km/1000m = 443 x 1000 km/s    (clear out the m, simplify)
then let's convert s  to hr.    443, 000 km/s = 443,000 km / s  x 3600s/hr = 443, 000/ 3600 km/hr = 4430/36 km/hr = 1594 km/h

1/ How far does a jogger run in 1.5 hours  if his average speed is 2.22 m/s ? in m and in km.
hint: speed = distance / time. First convert 1.5 hours into seconds.

2/ A bicycle has an average speed of 5mi/hr. How far it will travel in 2 hr ? in miles, in kilometer.
(see home page to find the conversion factor)

3/ convert 70 km per hour to miles per hour

4/ What units would have an appropriate size for measuring the average speed with which a blade of grass grows ? m/s ? cm/s ?  km/s ?

5/ A traveler covers a distance of 460 miles in a time of 8 hours. what is the average speed for this trip ?

6/ A traveler covers a distance of 1.8 km in a time of 30 minutes. what is the average speed in km/h ? hint: 60 minutes/1 hr =1

7/ A person in a hurry averages 62MPH on a trip covering a distance of 300 miles.
What time was required to travel that distance ?

8/ A hiker walks with an average speed of 1.2 m/s . What distance in km does the hiker travel in a time of 1 hour ?

9) light from the sun reaches EArth in 8.3 min. The velocity of light is 3 10⁸ m/s. How far is EArth from the Sun?
hint: convert minutes to seconds

10) you and your friend each drive 50km. You travel at 90km/h ; your friend travels at 95km/h. How ong will your friend wait for you at the end of the trip?
hint: you may want to concert to minutes.  Find the time for each person and take the difference.

11) you drive a car for 2h at 40km/h, then for another 2h at 60km/h
A) What is your average velocity ?
hint: Av velocity = total distance/total time
So first, find the distance coverd during the first 2h, then the distance covered into the next 2h, find the total distance and divide by the time.

12) A car is moving down a street at 55km/h. A child suddenly runs into the street. If it takes the driver 0.75s to react and apply the brakes, how many meters will the car moved before it begins to slow down.
hint: Find out the distance for a time = 0.75s. Convert 55km/h to m/s first.




10) Fill the table:

MPH = mi/h	km/h	m/s
20 MPH	_________	_________
?	64 km/h	_________
60MPH	__________	___________
__________	__________	36 m/s
__________	160 km/h	__________

11) Look at this map:




A) What is the average speed between Kingman and Phoenix ?
between Flagstaff and Phoenix  ?
hint: first find the distance . units MPH
B) trace the displacement vector from Kingman and Phoenix. Using a ruler, find the magnitude of the displacement and its direction/
hint: Find the scale of the picture. measure the distance in cm between Kingman and Flagstaff. that much of cm found is for 120 miles. Then measure the distance between Phoenix and Kingman with a ruler. (follow displacement vector). Convert to miles using the scale. To find This is your magnitude. the direction , use a protractor and find the angle between the displacement vector and the horizontal (a line between Kingman anf Flagstaff). so ds = _____ miles @ _____ degrees south of east.

DISTANCE VS TIME , use your graphing skill. The equation of a linear line is y = m x + b
m is the slope and m is the y-intercept.

PART IV: DISTANCE vs TIME, SPEED, USE your graphing skills, MORE PROBLEMS

1) Get graph a paper. 

The average speed of a car is 20mi/h
Fill the table :

x= time (hr)	y= distance (mi)
1	20
2	____________
3	____________
4	______________
5	____________
6	_______________

What is the relationship between the distance and the time ? d = ___ t (or y = ____ x)
This equation is called a ___________ relationship. The distance traveled is _________ to the time elapsed.
d is called the ______________ variable and t the ______________  variable.
Each time the time increases by 1 the distance increases by  _______  ?

graph distance versus time.  label the coordinates and don't forget the title. Find a good scale
for each of your axes. What is the slope of the line ?_____________________
 (slope= rise/run = change in y / change in x ).

Conclusion:
the average speed is the ratio of the distance covered (change in y) to the time elapsed (change in x).
The average speed is the __________ of the straight line in a position-time graph.

2) get a graph paper:
The following table records the distance covered by a toy car and the time elapsed.
A) plots the points and find 4 best fit lines to go through the points. you should find 4 parts in your motion
time = x    and distance = y
 

time (s)	distance (cm)
0	0
5	4.1
10	7.9
15	12.1
20	16.0
25	16.0
30	16.0
35	18.0
40	20.1
45	21.9
50	24.0
55	22.1
60	20.0

B)Describe the 4 parts of the motion.

C) Find the average speeds by computing the 4 slopes. (in cm/s)

3)  A cyclist maintains a constant velocity of + 5m/s (that is 5 m/s to the right. THe "+" is for the direction). At time t = 0.0 , the cyclist is 250m from point A.
 Plot a position-time graph of the cyclist 's location from point A, you can first build a x,y table.
x= time (s) and y = distance from point A (m)

time (s)	position from A (m)
0	250
10	_____
____	_____
----	-------
60	________

B. What  is the cyclist position from point A at 60.0 s ?
C. What is the slope ? its meaning ? What is the y-intercept ? its meaning ?

D. What is the relation between the position and the time ? (that is find the equation of the graph)
y = ____ x + ____ or ds (m) = _____ t (s) + _______


4) Both car A and car B leave school when a clock reads zero. Car A  travels at a constant speed
75km/h, and car B travels at a constant speed 85km/h.

A. Draw a position-time graph showing the motion of the car. Use the same graph, a different color for each line. Build a x,y table first for each car.

CAR A x = time(hr) y = position(km) 0 0 1 _____ 2 _______ 3 ____________ 4 __________ ----- __________

CAR B x = time(hr) y = position(km) 0 0 1 ________ 2 _____________ 3 ____________ 4 __________ --- ____________ ---- __________

B. Using your graph, How far are the 2 cars from school when the clock reads 2.0 h ? 
(when x = 2, find y)

C. Both cars passed a gas station 120km from the school. When did each car pass the
gas station ? calculate the times and show them on your graph.
hint: you should use your graph. trace a horizontal line at y = 120km and find tA and tB.

5) Graph and find the slope. The slope is the ___________. The equation of the graph is _____________
time (s)   position (m)
0                 0
1                60
2              120
3               180
4               240
5               300

6) The total distance a steel ball rolls down an incline at various times is given in the TABLE below.

distance versus time
x= time(s)	y= distance(m)
0	30
1	30
2	35
3	45
4	60
5	70

A) Draw a position-time graph of the motion of the ball. When setting up the axes, use five divisions for each 10m of travel on the y-axis.
Use five divisions for 1s on time for the x-axis

B) What type of curve is the line of the graph ? (first plot the dots and find your best fit curve. It is not a line !!!)

C) What distance has the ball rolled at the end of 2.2s ?

(hint: you can read the answer from your graph. Just find the intersection beween x = 2.2 and the curve.

If you want to use the TI. First plot the dots using STAT EDIT then STATPLOT then ZOOMSTAT.
But instead of finding the best fit line, you need to find the best fit parabola. A parabola has  a quadratic equation.
(y= ax2 + bx + C) the TI can find this equation of the graph for you by computing the coefficient a, b and c.
Here is how: STAT then CALC then scroll down to QUADREG   . you have your equation y = ax2 + bx + c
just plug in x =2.2 to get your y. Or you can enter the equation in Y then trace the graph on the top of the
scatterplots. Find the y for x=2.2. Use the up arrow to switch from te scatter plot to the curve)
HERE IS A MOVIE that shows you how to fit a parabola to a scatter plot and how to find the equation of the parabola
that best fits the data.

7) Members of a physics class stood 25m apart and used stopwatches to measure the time of a car driving down the highway passed each person. The data they compile are shown below:

X = time (s)	Y = position (m)
0	0
1.3	25
2.7	50
3.6	75
5.1	100
5.9	125
7.0	150
8.6	175
10.3	200

Using your graphing calculator to fit a line to a position-time graph of the data and to plot this line.
Find the equation of the best fit line: Y= ________ X + ___________



PART V: DISPLACEMENT VS TIME, VELOCITY

1) speed is a scalar and velocity is a vector. Velocity has
therefore a magnitude and a direction. speed tells you "how fast".
Velocity tells you "how fast and which direction"
A)  A car can moves at 20m/s , 60 degrees North of EAst. Can you represent that situation ?
hint: To represent the vector, use an arrow. The magnitude is 20m/s but you should scale it down  to draw it.
Like 5cm on your graph paper = is for  10m/s in reality .  The length of the arrow should be 10cm on your graph paper.
To find the direction, use a protractor. Your arrow should make an angle of  + 60 degrees with the horizontal.
The arrow is then pointing North of EAst/

B) What about 40km/s going South ?
hint: don't worry about km/s/ Just chose a scale like 1cm is for 10 km/s

C) 10km/s going 30 degrees North of West ?

D) 20cm/c going 30 degrees South of Earth ?

E) What about a velocity of - 50km/hr ? (the negative means to the left or westward ?

F) velocity of 60m/s ?


2) A race cars takes 2 runs. One toward the right. ds= + 604m and dt=2.19s
and one toward the left. ds = - 604m (that is @left) and dt = 2.22s.
Compute the average velocity for each run.
Vav = ds /dt    ds is the displacement.

3)

This graph shows the displacement (can be negative) of a car vs time.
Describe the 3 parts of the motion. For each part, find the velocity of the car (can be negative, the the car is going @ left
or @ west). Follow the steps:
PART1    0< time <2s (I help you forthis part)
The car is moving __________________________ ( not moving ? moving@ left ? moving @ right ? at  a constant speed ? accelerating? )
Every second , the car covers a displacement of _______ m (negative is for @left)
The slope is ______ m/s (the slope is negative. slope = rise/run or easier :  for a run = 1 what is the rise  , how many miles per second?)
The velocity V = ________ @ _______ (magnitude = slope, direction is given by the sign)
PART2: 2< time <4s
The car is ____________________
The line is horizontal, the slope = _____ m/s
The velocity = ____________
PART3: 3< time <6s
The car is ____________________  @ _________
Every second, the car covers _________ m
The slope is __________
the velocity is ____ @ _____________

4) From the position-time graph (that is displacement vs time) belowm
describe the motion of a track runner.



between 0<t<40s      the velocity is V = ____m/s @ ________
and _________

between  40<t<70s _________________________

between 90<t<100s the velocity is V = ______m/s  @ _______ or - _______m/s

5) Sketch position-versus-time graph. Sketch means and-drown graph. You don't need to include any numbers.
Just sketch the graph of the motion of:
A) A student walks to the bus stop , waits for the bus, then rides to campus. Assume that all the motion is a long a straight line
hint: there are 3 parts in the motion

B) As tudent walks slowly to the bus stop, realizes he forgot his Physics lab that is due, and quickly back  home to get it.
hint: 2 parts in the motion. Since he walks back, the position decreases (he get closer to 0 = home) . And het gets closer and a higher rate. (slope negative = downhill but larger magnitude)

6) the position versus time graph below shows the position of an object moving in a straight line for 12s.



A) What is the position of the object at 2s, 6s, and 10s after the start of the motion?
At 2s: ____________

At 6s: ___________

At 10s: ____________

B) What is the object's velocity during the first 4s of motion?

C) What is the object's velocity during the interval from 4s to 6s ?

D) What is the object's velocity uring the four seconds from 6s to 10s ?

E) What is the object's velocity during the final 2 seconds from t10s and t  12s ?

F) Draw a motion diagram below to represent the entire 12s of motion.
That is draw a graph velocity versus time. This is more advanced stuff.
Try but we will go over in class together.






PART VI: VELOCITY VERSUS TIME, HOW TO FIND THE DISPLACEMENT

1) A) What is the area of a rectangle ?

B) What is the relationship between average velocity V and displacement  ds ? ds = ______ x _________
hint: think time t
 So  displacement ds = velocity V x time elapsed t
time elapsed t  should be written dt but I will call it t instead.

C) observe the graph velocity versus time. The velocity is given by a speedometer.
It is called the instantaneous velocity  (we will see that lateR).  



between the time 0 < t < 20s, the velocity is about V = _________ m/s.
The time elapsed is t = ________s.
The displacement of the car = ____________m.
hint: ds = V t
The area of the rectangle below the graph, between 0 and 20 s = _____________
So displacement = area under the velocity vs time graph !

This will be true even if the area is not a rectangle !

D) Find the displacement for the 3 other time intervals. Remember, displacement can be negative.(in which case, the car is moving @ left )

For 20s<t<30s      ds = __________m
For 30s< t < 50s     ds = __________m
For 50s<t<60s          ds = __________m


We can find the displacement this way even when  the graph velocity vs time
does not involve any rectangles. The process is more complicated as
you need you know how to integrate over time to find the area under a
more complex curve.  (in calculus 2)

2)

A) graph :

x=time (s)      y= velocity (m/s)
0                      0
1                       20
2                       40
3                       60
4                       80
5                      100

B) WHat is slope of the line ? a = ____ m/s/s
Can you find the equation of the line ? y = ____ x or velocity (m/s) = _____ x   time (s)

What does the slope represent ? (observe the units, m/s/s)  

That means  every second, the car increases its speed by ______ m/s

C) Can you find the displacement covered after 1 s ? after 2s ?
(hint: find the area below the line in the time interval 0 and 1s then between 0 and 2 s)

D)                                                                   
Fill the table :
time (s) elapsed       displacement (m) covered
0                            ______________
1                            ______________
2                            ______________
3                            ______________
4                            ______________
5                            ______________
                                                                          
(hint :you can use the graph and compute the areas or you can use the formula d = V t )

INSTANTANEOUS VELOCITY

A speedometer measures the instantaneous velocity.  
It tells us how fast we go at a given instant in time. It tells us
little how long it will take to travel several miles. Unlike the average velocity.
We can find the instantaneous velocity  by computing the average velocity over a short time interval in which the velocity 
does not change appreciably.
Vinst = (displacement)  / t   with t very very small. . If you take calculus, you will find out that Vinst is the
derivative of the ratio distance / time. 





1) Variations in instantaneous speed for a portion of a trip.  Describe the trip.