PART 1: AVERAGE ACCELERATION

1) average acceleration = change in velocity / change in time.  (a_av = dv / dt)
Whenever velocity is changing - in either magnitude or direction - the motion is said to be accelerated.
In a car, what are you using to change the velocity ? (3 things)

Your car can get from 0 to 60mi/hr in 9.2 s.
The average acceleration is __________mi/hr/s
The speed increases by ____________mi/hr   in each second of the trip/

2) READ notation:
velocity, acceleration and, displacement are vectors. They have a magnitude (how much), units (of what) and,
a direction (which way).
Initial velocity will be noted Vo (for V at t=0s) or Vi (i for  initial). V1 is another equivalent notation.

the acceleration will be noted a. you will see that a can be negative if you are slowing down. (but moving to the right)
or if you are speeding up, but moving to the left.

The final velocity will be noted V or sometimes V2 or Vf (f for final)

t is the time elapsed during the motion we are studying.

x is the position on the number line. The position can be positive or negative (if you are on the left side of 0, on the number line)
and of course, x - xo is the displacement, the change in position. For most of the problem, xo=0.

1) Suppose a car is traveling at 12 m/s and increases its speed uniformly to 30m/s, taking 15 s
to do so. Uniformly means the acceleration is constant. What is the value of the acceleration ?

2) What is the acceleration of a rocket in outer space that takes 5.0s to increases its speed from 1,240 m/s to 1,300 m/s ?

3) How much does the speed of a car increases if it accelerates uniformly at 2.5 m/s² for 5s ?

4) A car is going 8.0 m/s on an access road in a highway, and then accelerates at 1.8 m/s² for 7.2 s.
How fast is it then going ?

5) A plane is taking off. at t= 0s Vi =0 and and t = 29s v = + 260km/h
Find the average acceleration in km/h/s.


PART 2 ACCELERATION IS A VECTOR

1) Observe the following picture:

What is going on ? __________________________
If V1 = + 10 m/s  and V2 = + 30 m/s
Find ΔV = V2 - V1 = ______ (don't forget the sign, it shows the direction)
a = ΔV/ΔT  a is a vector. a It has the same direction than ΔV
If the car changes its velocity in 5 s find a = ________  (don't forget the sign)


2) Observe the following situation:


What is going on ? Predict the sign of V2 - V1 _________________________________________
If V1 = + 30m/s (the direction East is represented by a positive sign )
V2 = + 10 m/s
Find V2 - V1 =  ________  (don't forget the sign. can be negative)
if ΔT = 2s (time elapsed between the 2 situations)
Find a = _________  !! a is a vector. Use the + or - signs to show the direction.
which way points a ? Can you trace a on the diagram ?

SO:
If a car or an object is slowing down,  the vector velocity and the vector acceleration have opposite direction and,
 therefore opposite signs. So if  an object is moving to the right and is slowing down, 
velocity has a positive sign (moving to the right) but  acceleration has a negative sign and is pointing to the left.

V2 < V1

3) A drag racer crosses the finish line, and the driver applies the brakes to slow down.
The brakes are initially applied when to=9.0s and the car's velocity is V1 = + 28m/s.
When t = 12.0 s, the velocity has been reduced to V2 = +13m/s.
A) What is the average acceleration of the dragster ? (acceleration is a vector, it can be + or -)
B) Can you trace the vectors (V1, V2 and a) ? scale : 2 cm = 10 m/s (use a graph paper)


4) The driver of a car steps on the brakes, and the velocity drops by 20m/s (V2 - V2) due east in a time
of 2.0 s. WHat is the acceleration ? (acceleration is a vector. show the direction)
a = _____ @ _______

5) A runner moving with an initial velocity of V1= 9.0 m/s slows down at a constant rate of a = - 1.5 m/s/s
over a period of 2 seconds. What is her velocity at the end of this time  ? after 1 second ?
(the acceleration can be negative (you are slowing down) but the velocity be positive (you are still
moving to the right even if slower)

6) Starting from rest (meaning V1 = 0 )and moving in a straight line, a runner achieves a velocity of V2= 7 m/s in a time of 2s.
What is the average acceleration of the runner ? (if you are speeding up the acceleration is positive)

PART 3: INSTANTANEOUS ACCELERATION

1) Read
Instantaneous acceleration Is the rate of change of velocity at given time.
It is computed by finding the average acceleration for a very short
time interval during which the acceleration does not change appreciably.
Here we will only deal with a constant acceleration.

You stomach is our detector of instantaneous acceleration: It will senses the downward acceleration of an elevator.


PART4 : GRAPHING MOTION AND THE BIG FIVE
(5 equations to describe motion with a constant acceleration along a straight line)

try without hints. unless you get "confused"
1) The table gives some data for a Ferrari racing car at the start of a Grand Prix race:

speed ( m/s)	0	10	20	30	40	50	60
time (s)	0	1	2	3	4	5	6
distance (m)	____	____	___	____	___	___	___

A) Plot a speed-time graph for the car (x = time and y = speed)
B) Find the slope. What does the slope represent ? pick one : speed ? acceleration? displacement?
The acceleration of the car a = __________m/s/s
C) Can you find the distance covered by the car at a given time ? complete the table above.
 remember distance = area under the curve speed vs time  = area of a triangle.
hints: for example. for t =2 area = 0.5 (speed) (time) = 0.5 (20) ( 2) = 20m

D) so at a given time t and given the velocity V,
you can find the distance covered by the car using the formula:

d = _______  (use in your formula the time elapsed t and the acceleration a.)

hint: step 1: distance = area of a triangle of base t and height V.    d = _________ (step1 formula)
step2:  a = V /t   or V = at. substitute V = at in the step1 formula.    

E) Let's find the average velocity of the car 2 different ways:
i) average velocity = total distance / total time.   so find the total distance and the total time
hint: the total time is 6s. The total distance was found in the previous table. For t = 6s the total distance = _____ m.

ii) Look at your graph. Could  you have found the average velocity using your graph ? how?
Could you have found the average velocity by using the table ? how ? 
hint: the average is in the middle  (it is the median) of all the values. this is because the acceleration is constant.

F) Plot the distance vs time . What kind of graph is it? It is ok to use the TI as I showed you.

2) Read
The previous problem has shown you:
Vav = (V1 + V2) /2    where V1 is the initial speed (0 m/s) and V2 is the final speed (60m/s)
and d = 0.5 a t² when V1 =0 where d is the total displacement, a the constant acceleation and t the time elapsed
Get index cards. Write (you can bring them for the test/quiz)

1) a =(V2 - V1) /t

or 2) V2 = V1 + at

3) d = 0.5 a t²   if Vo = 0

4) Vav = (V1 + V2)  /2


2) Suppose now that the car has an initial speed of 10m/s at t=0s  Vo= 10 m/s
A)Fill  the first row of the table. Keep the same acceleration. ( a = 10m/s²)

speed ( m/s)	10	____	___	___	___	___	___
time (s)	0	1	2	3	4	5	6
distance (m)	___	____	____	___	____	____	____

B) plot the graph speed vs time. (don't forget title ...)
y= speed and x = time.
What is the y-intercept ? V1 = ________ m/s
What does it represent ? ________________________
Find the slope a = ________ m/s/s.   what does it represent ? _______________________

C) Can you find the distance covered at any time by computing the area under the line? (think geometry)
Fill the third row of the table.
hint:the area under the line = area of the triangle + area of the rectangle. Fill the table.
example:for  t = 1s , distance d = 0.5 (1) ( 10) + (1)(10) =5 + 10 =15m, for t= 2s you should find d = 40m ...
the area under the line  = area of a triangle + area of a rectangle.

D) You just have found another equation of motion:
When the initial speed is non zero (V1 = 10m/s) : total distance  = area of the rectangle +  area of the triangle
 so total distance d = _____ + __________ (in your formula use t, V1, a )

hint:  d = 0.5 (height of triangle = at) x base (t))  +  ( width of rectangle (t) x height (V1 )  )

Add this formula to your index card

E) What is the AVERAGE speed ?
i) use the formula Vav  (V1 + V2) /2
ii) use your table and find the the median of 10, 20, 30 , 40 , 50 , 60 , 70

F) graph distance vs time. What kind of graph is that. It is ok to use the TI, Show me the graph. (or other tool)/

3) Read THe graph should have looked like this one:



The displacement is the area under the graph velocity versus time.
You can break the area into 2 parts. A rectangle (displacement covered if the car does not
accelerate but keeps the same velocity of 10 m/s) and a triangle (the additional displacement due
to the acceleration. More distance is covered because the car accelerates.

4) play with this applet.
Observe the 3 graphs : acceleration vs time, distance vs time and speed vs time.
simulation: acceleration of a car.

5) THE BIG FIVE (write them on index cards)
Initial velocity will be noted V1 (for V at t=0s)  
the acceleration will be noted a. you will see that a can be negative if you are slowing down. (but moving to the right)
or if you are accelerating but moving to the left.
The final velocity will be noted  V2 
t is the time elapsed during the motion we are studying.
d is the displacement. d can be negative. 5 blocks to the right + 8 block to the left = displacement of -3

So far you know: (on your index cards)
average velocity = total displacement / time elapsed  or Vav = d / t

or Vav = (V1 + V2 ) /2           (acceleration is constant)

a = (V2 - V1) /t     or acceleration = change in velocity / time elapsed                 

or V2 = V1 + at
.
d = V1 t + at² / 2   If V1 = 0 then d = at² / 2   also   if  a = 0  then d = V1 t  = V t    the velocity stays constant.

Then at last:   V2² = V1² + 2ad          
You didn't demonstrate this last one, but you could using algebra and playing with the previous equations
Playing with the equations:
d = V1 t + at²/2      
a = (V2 - V1 )/t
 d = Vav t with Vav = (V1 + V2) /2

DO LAB2 reaction time
DO LAB1 acceleration

6)
The 5  fundamental quantities used to decribe an uniformly accelerated motion along a straight line are:
V1, V2 , ___ , _____ , _____

These five quantities are related by a group of five equations that we call the big five/
You can use these equations of motion ONLY IF tHE ACCELERATION IS CONSTANT.
Only if the velocity increases by the same amout every second. (direction and magnitude).
We say the object is uniformly accelerated. Here we are only dealing with objects moving along straight line. So there is no change in direction. Only a constant change in magnitude. You will deal with circular motion next year.
(For example satellites move along circle at a constant speed but they always change their direction. The change in direction is cosntant.
We call that a uniform circular motion)
Can you write these five equations ? (see  question 4) )

		variable that's missing
five 1	(displacement, a vector) d =	a acceleration
five 2	(velocity, a vector) V =	d displacement
five 3	d =	V (Vf)  final speed
five 4	d =	Vo initial speed
five 5	(final speed squared) V2 =	T time elapsed

7) a jet is taking off from the deck of an aircraft carrier. starting from rest (V1 = 0m/s) , the jet is catapulted
with a constant acceleration of a = + 31 m/s² along a straight line and reaches a velocity  of V2= 62 m/s.
Find the displacement d of the jet. (play with the big five)


8) a speedboat has a constant acceleration of a= 2 m/s² . If the initial velocity of the boat is V1= 6 m/s,
calculate its displacement after t = 8.0s

9) How long does it take a car, starting from rest, to travel d= 240m if it its acceleration is a = 1.90 m/s² ?

10) What is the acceleration of the rocket that accelerates uniformly from rest and travels
d= 650m in the first 12 s ?

11) An object with an initial velocity of V1= 4 m/s moves along a straight  line under constant acceleration.
3 seconds later, its velocity is V2- 14 m/s .   How far did it travel during this time ? (find d)


12) A car's that's initially traveling at V1= 10m/s accelerates uniformly  for 4 seconds at a rate of 2 m/s²
in straight line/ How far the car travel during this time ? (d )




Try:
13) A car going V1= 22 m/s (initial velocity )  has its brakes jammed on and leaves skid marks d = 45m long.
what was its acceleration ? (the car stops)
hints: Vi = 22m/s and V2 = 0 m/s  the acceleration is negative

You could have taken a short cut and use the formula:
V2² = V1² + 2ad   
14)
How fast is a car going if it starts at rest and accelerates uniformly a= at 2.8 m/s² while traveling 220 m ?



15)
QUIZ or homework

16) A ball initially at rest (V1=0) rolls down a hill with an acceleration of 3.3m/s/s. If it accelerates for 7.5s, how far will it move?

17) A plane starting at rest at the south end of a runway undergoes a constant acceleration of 1.6 m/s/s for a distance of 1600m before takeoff.
A) What is the plane's velocity at takeoff ?
B) What is the time required for take off ?

18) A bus slows down uniformly from 75 km/h (21m/s) to 0.0 km/h in 220m. How long does it take to stop /

19) A car accelerates from rest at -3m/s/s
A) what is the velocity at the end of 5s ?
B) What is the displacement after 5s ?

20) A car starts from rest and travels for 5s with a uniform acceleration of 1.5 m/s/s. The driver then applies the brakes, causing a uniform acceleration of -2m/s/s. If the brake are applied for 3s, how and how far has it gone from its start ?

21) A boy sledding down a hill accelerates at 1.40m/s/s/ If he started from rest,
 how does it take him to reach s speed of 30m/s ?

PART5: REVIEW - MORE GRAPHING PROBLEMS / conceptual questions

1) REVIEW

A) The rate at wich the velocity changes is called ___________________
B) In terms of speed and direction, in what ways can an object accelerate ?
C) Because acceleration is a quantity that has both magnitude and direction, it is a ____________
D) Ignoring air resistance, a rock in free fall will have a velocity of ______________ after 4 seconds. (a = - 9.8m/s/s )
starting from rest.
E) A horse on a carousel that is moving at a constant speed is accelerating because ________________________
F) What is meant by "constant acceleration "
G) Write the equation used to claculate the acceleration of an object

2) REVIEW

GRAPH A

GRAPH B

A) Graph A represents the motion of a downhill skier. How fast was the skier moving after traveling down the hill for 2.5 seconds ?

B) Find the acceleration of the skier.

C) GRAPH B represents the motion of a mountain biker. What is the biker's speed at times of 10s and 20s ?

D) Find the acceleration of the biker between 0 and 10s, then between 10s and 20s ?
(!!! slope is negative)

E) why the acceleration is negative during the second time range ?

3) REVIEW
The measure of how fast a velocity is changing at a specific instant is known as _________________________

4) complete the sentences
- An expression for _____________________ is (Vf - Vi) /t
- A quantity that has both magnitude and direction is called  a ____________-
- The total distance traveled divided by the total time is ________________
- If a speed-time graph is a line, the acceleration is _________________ and the relationship between speed and time is _________.
- Common units for ____________ includes m/s
- In order to acurately and completely describe the motion of an object, a ________ of _____________ is necessary.
- You can determine ____________ by measuring the length of the actual path between 2 points in space
- The graph distance vs time , for a non zero but constant acceleration, is a curve. What kind of curve ? ___________________-
We say the relationship between distance and time is non _________ but qua................
- The word for the rate at which velocity changes is ___________
- The word for how far an object has moved _____________

5) The graph below who the speed of a free-falling body versus time. Free falling means we neglect air resistance.



A) What is the relationship between speed s and time t?  (use the graph and its equation)

B) what does the slope of the graph represent (the physical quantity)? ___________ due to ______________
The units o the slope are __________
The slope tells you that every second, the speed increases by _______________

C) Is the following sentence true or false ?
In real life, air resistance has no effect on the acceleration of a falling body.
(think of dropping your Physics book and a paper)

D) An approximate value of the acceleration of an object in free-fall is _____ (your graph)
An acurate value of acceleration of an object in free fall is ____
The symbol (a letter) used to represent acceleration due to gravity _____

6) Let's study the distance covered by a freely falling body versus time. We neglect air resistance.



A) Is this statement true (detail your answer, for every second of the fall find the distance covered)
For each, second of free-fall, an object falls a greater distance than it did the previous second.

B) the graph is a _________________, the relationship is ________________, that means that the distance
is proportional to the _________ of the time.

C) Can you find the equation of the graph ? It will give you the relationship between distance and time, when an object falls on EArth. (air resistance is neglected).
(hint: the acceleration of a free-falling body is the acceleration due to gravity. You have the acceleration, you should know the relationship between distance and acceleration using your index cards. see the big five).

D) Is the speed constant ? Using the graph, can you  think of a way to estimate the speed at a given time ?
(hint: remember ? speed = slope of distance vs graph)

E) As the times goes by the speed __________ at a rate of _________ per second.






LABS

2) materials : incline planes, stop watch, steel ball, block, masking tape, pencil ..
print out:
part1     part2    

3) materials: protractor, string, small mass
part 1    part2