free-fall no air resistance
free-fall air resistance
source: a great website tutorial

a nice animation to review constant acceleration motion
animation to see the relation velocity as vector and acceleration as a vector

watch a movie about Galileo and the study of motion along incline plane.
By diluting the acceleration due to gravity, we was able to find the relations: V = gt and d = 0.5gt²

PART 1: FREE-FALL
record for free-fall falling.

0)Read:  If you drop a rock and a feather in an air-filled tube, the rock reach the bottom first.
This is due to the presence of air resistance. In an evacuated tube, in absence of air
resistance, both the rock and the feather have the same acceleration. They will reach the ground
at the same time. the acceleration due to gravity us denoted by the symbol g. g is a vector .
g = 9.8 m/s @ down or g = - 9.8 m/s² (surface of the Earth)

Free-Fall is an example of an uniformly accelerated motion with a = - 9.8m/s²
(That is the acceleration is constant = each second, the object increases its speed by 9.8m/s)

^{When doing problem, the first thing to do is to decide of a positive direction (let's take up as positive)
and a negative direction ( let's take down as negative). The origin is usually chosen at the location
the object is just before you throw it  or drop it.}

When you use the equations of motion, you have to pay close attention to the sign of :
- the acceleration g pointing down. So a = -9.8m/s/s 
- the initial velocity noted Vi or Vo or V1. positive if the object moves upward at t=0s.
negative if it is falling at t=0s.
- the final velocity V or Vf or V2. moving up = positive. moving down = negative.
- the displacement (change in position along the y-axis)  d of the object. positive if the object moved toward the positive numbers on the number line (moving up). negative if object moved toward the negative number (moving down).
(displacement is a vector. tail = starting point, head = ending point)

Exercise:

0) review questions:
A)What does the slope of a position-time graph indicate ?
B) What quantity is represented by the area under a velocity-time curve ?
C) What does the slope of a velocity-time indicate ?
D) If a velocity-time curve is a straight line parallel to the x-axis, what can be said about
the acceleration ?

1) The following table gives the positions and velocities of a ball at the end
of each second of free fall from rest.

time (s)	position (m)	Velocity (m/s)
0.0	0	0
1	-4.9	-9.8
2	-19.6	-19.6
3	-44.1	-29.4
4	-78.4	-39.2
5	-122.5	-49

A) Use the data to plot a velocity-time graph. ( meaning x = time and y = velocity)
 Find the slope. (find the best fit line going through the origin)
Slope = ________________ (don't forget the units). What that represents ? __________

you can use your TI. clear Y= , set STATPLOT on,  use STAT EDIT to enter xs in L1 and ys in L2,
use ZOOMSTAT to plot, decide what kind of graph this is, use STAT CALC to find the best fit line (or parabola
if it is a parabola), write down your equation. enter your equation in Y=, GRAPH, the graph of
your equation should connect the dot.

B) Use the data in the table to plot a position-time graph (meaning x=time and position = y)
The easiest way is you use your TI. see above for steps.
What kind of curve is this ? _____________ The position (the y) varies as the square of the _______ (the x).
Find the equation of the graph using the TI. (see above , use STAT CALC QUADratic regression)
you find y = _______ x² or d = ______ t² (since x = t)

Here the acceleration = a = ___________ (sign please)
Does your equation agrees with the formula d = 1/2 a t² (no initial velocity, check index card)

C) Find the displacement of the ball when t = 2.5s. (use your equation)

PART2: Throwing a ball upward

REmember velocity, displacement and acceleration are vectors. their sign tells
the direction they are pointing to. Positive = pointing up. negative = pointing down.

observe the picture:


1) Look at the picture. (from The physics of everyday phenomena, Mc Graw Hill).
Use the initial conditions : Vo = 20m/s and g = - 10m/s ²  
At the top of the path, the ball has a velocity = ________ and an acceleration = ________
Actually, t=0, t=1s, t=2s, t=3s, t=4s the acceleration = ___________
Because the ball is always acted upon by the force of ___________


2) A) Let's study the motion of ball thrown up  in 1). (Vo= + 20/s). we neglect air resistance.
Using one of the Big Five, can you find d (displacement of the ball )
and V (velocity of the ball) for the times shown ? they can be negative/ 
(with Vi = 20m/s, a = -10m/s/s).  SEE TABLE NOW. then use hint.

hint: The easiest way is to use your TI. Enter the equations into Y= (see upper left TI )
enter Y1 = 0.5 a x²  ( substitute a = - 10 before ,  so Y1 = -5 x² of course)    
Y2 = Vo + ax (substitute a = -10 and Vo = 20 before, so Y2 = 20 - 10x)
then use TABLE (TI)  to find the displacements and the velocity for a given time.
(TABLESET has to be on ASK)

time (s)	displacement (m) d= 0.5 a t2 a = -10m/s/s	velocity (m/s) V= Vo + at Vo = 20m/s a = -10m/s/s
0	_____	_____
1	_____	_____
1.5	_____	_____
2	_____	_____
2.5	_____	_____
3	_____	_____
3.5	_____	_____
4	_____	_____

B) do you find the same values as in 1) ?

C) note that when the ball reaches its highest point, its VELOCITY IS __________. very important. INDEX CARD.
something to remember. Note that when there is no displacement the velocity can be +_______m/s or - _______m/s
but the magnitude is the same.
 (we neglect air resistance, energy is conserved, the motion up is symetric with the motion down)

D) how long does it take for the ball to go up and to go down ?
is it the same time? remember that. time up = time down . INDEX CARD.

3) A ball is thrown straight up into the air at an initial velocity of 25m/s.
A) Create a table showing the ball 's position,
velocity, and acceleration each second for the first 5.00s of its motion.

t (s)	y (m)	v(m/s)	a (m/s/s)
0	_____	_____	_____
1	_____	_____	_____
2	_____	_____	_____
2.5	_____	_____	_____
3	_____	_____	_____
4	_____	_____	_____
5	_____	_____	_____

hint: you can use your TI. The acceleration = acceleration due to gravity
see problem before for steps. enter the equation for y(m) into Y1 and the equation for v into Y2.

B) Find the ball's time, position, velocity, and acceleration to the top of its flight.
hint: V = 0 at the highest point.

C) Can you see some king of symmetry in the table?

D) how long does it take for the ball to go up and to go down ?
is it the same time? remember that. time up = time down . INDEX CARD.

4) 
The acceleration of an object in free fall (in a vacuum) does not depend on the nature
of the object. It depends only on where the objects happens to be.
On earth g is about 9.8 m/s². 3.3 m/s2 on Mars, 26.6 m/s² on Jupiter, and 1.67 m/s2 on the moon.
It increases with latitudes on earth. It is 9.83 m/s² at the poles and 9.78 m/s² at the equator.
It also decreases with altitude. 9.79 m/s² at altitude 10 miles.


souvenir of class 2008/2009 10 th grade, section 2 , 2/5/09

5) A high-wire artist missteps and falls 9.2m  to the ground. What her speed on landing ?
(hint: use one of the big five. d= - 9.2, a = -10m/s/s, Vi=0   the displacement is negative because final position = 0 and initial position = 9.2
so displacement = final - initial = 0 - 9.2 = -9.2m)

6) A cat falls out of a tree, dropping 16 m to the ground. How long is the cat in the air ?
(hint: d = -16m, Vi = 0 )

7) You drop a rock from a bridge, and it hits the water 2.3 s later. Find (a) the height of the bridge ?
(b) the velocity of the rock when it hits

8) you throw a baseball straight up, and it leaves your hands at 15m/s.
What is its velocity 2.0 s later ?
down is negative so g = -10 m/s and Vi = +15m/s  
Don't forget the sign when plugging the acceleration into the equations.

9) A ball is thrown vertically upward with an initial velocity of 98.5 ft/s (Vi = 98.5).
you are working with feet and second so the acceleration is now a = - 32 feet/s/s  (32ft/s/s = 9.8m/s/s)
A) how high the ball goes ?
hint: at the highest point Vf = 0. use Vf² = Vi² + 2ad, one way.

B) how much time does it take to reach its maximum ?

C) what is the total time the ball is in the air before striking the ground ?
hint: it takes the same time to go all the way up or all the way down.


10) Just as car A is starting up (that is Vi_A = 0), it is passed  by car B.
Car B travels with a constant velocity of 10m/s (that is acceleration of B :  a_B = 0 and V = Vi_B = 10m/s)
While car A accelerates with a constant acceleration of 4.5 m/s² ( that is aA = 4.5m/s/s) 
A) Find the equation distance vs time for car A
d_A = ______ t²       ( hint: with a = 4.5 m/s/s and Vi = 0)

Find the equation distance vs time for car B
d_B = _______t       (hint  : a = 0, Vi = 10m/s)

B)  Compute the distance traveled by each car for times of 1s, 2s, 3s, and 4s
hint: use the TI. enter d_A in Y1 and d_B in Y2, use TABLE to find the distances.

time	distance car A	distance car B
1	____________	____________
2	____________	___________
3	__________	_____________
4	____________	_____________
5	___________	_____________

C) For each car plot, on the same graph, distance vs time.  Just sketch it. You should see when car A pass car B.

D) Using the graph, find at what time, approximately, does car A overtake car B ?
hint: Find the coordinate of the point of intersection (x,y) of the 2 graphs.  x is the time.
I can show you how to do it with your TI but might be faster with a graph paper. more advanced stuff.

D) Can you think of another method to do that ? Can you show the other method ?
hint; use Algebra. if the distance is the same, d_A and d_B are the same d_A = d_B). solve for t.

11) A ball is thrown straight upward with an initial velocity of 16m/s (Vi = 16m/s). use g =  - 10m/s².
a) What is its velocity at the high point in its motion? 
hint: check index card, highest point the velocity is _____________

b) How much time is required to reach the high point?
hint: plug the velocity found in a) in the equation: a = (V- Vi ) /t

c) How high above its starting point is the ball at its high point ?
hint: plug the time found in b) in the equation d = 0.5 a t² + Vi  t    (a = -10 and Vi = 16)

d) How high above its starting point is the ball 2 seconds after it is released ?
hint:same as above


12) An astronaut drops a feather from 1.2m above the surface of the moon. (d= -1.2m)
 If the acceleration of gravity on the moon is 1.62 m/s² (a=-1.622) downward, how long does it take the feather to hit the moon's surface ?

13) A stone falls freely from rest (Vi = 0m/s) for 8.0s.
A. calculate the stone's velocity after 8.0s
B) What is the stone's displacement during this time ?

14) A bag is dropped from a hovering helicopter.
What the bag has fallen 2.0s.
A) what is the bag velocity ?
B) How far has the bag fallen ?

15) A ball thrown vertically upward is caught by the thrower after 5s.
A) Find the initial velocity of the ball. try without hint.

hint: solve for Vi. a = -10. you know that it takes 5s for the ball to go up and down. so 2.5 s to go up and reach V = 0
at its  highest point.

B) Find the maximum height it reaches

16) A small sandbag is dropped from rest from a hovering hot-air balloon.
A) after 2 s, what is the velocity of the sandbag ?
B) After 2s, how far below the hot-air balloon is the sand bag ?


17) You jump from a 10m diving plank into a pool.
The function height vs time is y = - 4.9 t² + 10   The parameters of the quadratic function (4.9 and 10 ) tells you
about the initial conditions: acceleration is -9.8  (2x4.9), there is no initial speed, the initial height is 10m

Find the function that model the height of a diver jumping from a 10m diving plank on the moon (a = 1.64m/s/s).
y = _______________

Find in each case , how long it takes for the diver to reach the water. (solve for t if y = 0 ).
Find the ratio between the times.


17) EXTRA CREDITS

  Do you think this man will go faster with 1 ton mass ?
PART 3: (EXTRA CREDITS)  USING THE QUADRATIC FORMULA to solve for the time elapsed  - EXTRA CREDITS
TRY WITHOUT HINT

0) -You can use your TI to solve a quadratic equation like Ax²  + Bx + C = 0
that is finding the values x that will make the equation equal to zero.
In Physics, you are solving for the time. So you are only interested into
the positive solution. Here is 2 ways:

A)  you can use the equation solver. Let's try to find the solution of
x² -2x - 15 = 0. To use the equation solver, you will need to give a guess.
let's make x= 1 as a guess. watch the movie.
(we are not interested in the other negative solution)


B) You can also use the quadratic equation to solve and use your TI to solve the equation.
let's solve x² -2x - 15 = 0
A = 1   B = -2 and C = -15
you are going to use x = (-B + sqrt ( B² - 4AC) )/ ( 2A)
watch the movie   x=5 is the solution !

TRY YOUR NEW SKILLS:

1) The height of a toy rocket launched straight up with an initial Vo  is given by the function
f(t) = 48t - 16t²     time is in seconds but f(t) = vertical position y  is in feet.
Understand that f(t) represent the vertical position y. we are computing distances in feet so the acceleration is 32ft/s/s
(instead of 10m/s/s same thing)

A) you know that  the position of an object (y) , moving with a constant acceleration (a) is given by
one equation of motion:
y = 0.5 a t² + Vot + yo   (one of the big five).  Vo is the initial velocity, a the acceleration and ho the
initial position. Compare to the equation given f(t) = 48t - 16t²   (or y = - 16t² + 48t )
and find Vo = ________ ft/s   ,   a = __________ ft/s/s, ho = _______ feet.

B) graph the function for t = 0, 0.5, 1, 1.5, 2, 2.5, and 3.
you can use the values table:
(you can use your TI to find the f(t) using TABLE here is a short movie to find out how)
steps: make sure your STATPLOT is on, clear Y=, STAT EDIT, ZOOMSTAT

t	f(t)
0	___
0.5	____
1	_____
1.5	_____
2	______
2.5	______
3	______

What kind of curve is that ?

C) When is the rocket at his highest point ymax? What is its height ymax? (according to your graph)

D) Check what you find in C using one other equation of motion.
hint: check using the equation V = Vo + a t    
At the highest point V = 0. you know Vo = 48ft/s  and you know a = - 32 ft/s/s
solve for t=tmax (time to reach the highest point) and then solve for h(t) = ymax (the max height)
by plugging tmax  into h(t). Do you get the same thing?

E) How many seconds does it take for the rocket to land ?
hint: When the rocket lands y = h(t) = 0. (no more vertical elevation)
So you need to solve 0= 48t - 16t²
You can use your equation solver with your TI (use t=5 as a guess). , or you can use your math skill.
There are 2 solutions but only one makes sense. (t=0 does not make sense)

FOR THE COMING problems use the equation of motion:  y = 0.5 at² + Vot + yo instead of  d = 0.5 at² + Vot .
y is the final position (y-component or ordinate). yo is the initial position (along vertical)
It is the same thing because y - yo is the displacement d.

2)
Suppose that a flare is launched upward with an initial velocity of Vo = 80ft/s and from a a height of yo = 224ft.
(the acceleration due to gravity in feet/s/s is a= - 32 )
A) Its height y, in feet, after t seconds is given by:  f(t)  = y =   _________________

(hint: use the equation of motion y = 0.5 at² + Vot + yo    y is the final ordinate (y-coordinate ) and yo is the initial ordinate (y-coordinate)
we modified the equation d = 0.5 at² + Vot slightly to y = yo + Vot + 0.5at²   it is the same thing because y - yo is the displacement d.

B) After how long will the flare reach the ground ?

hint: you need to solve y = 0.
you can use the quadratic formula ( - B + sqrt(B² - 4AC) ) (2A) and your TI (see movie above)
or you can use the equation solver with your TI (the easiest way. guess = 3 )

3) Jesse is tied up to one end of a 40m elasticized (bungee) cord. The other end of the cord is tied to the middle of a train trestle.
 If Jesse jumps off the bridge, for how long will he fall before the cord begins to stretch ?

hint: you know the equation of motion  y = 0.5 a t² + Vot + yo. But suppose yo = 0 because you place the origin at the level of the level of the train,  Vo = 0m/s because he was at rest before jumping and a = - 10m/s/s.  
When the cord starts to stretch, the jumper has covered y = -40m. So solve :
y = 0.5 a t² + Vot + yo   with yo=0, Vo =0, y = -40 and a = -10

4) A  baton twirler tosses a baton into the air. The baton leaves the twirler's hand 6 feet above (yo= 6)  the ground and has an initial velocity of 45 feet per second (Vo=45). The twirler catches the baton when it falls back to a height of 5 feet (y=5).
For how long is the baton in the air ? suppose a = - 32ft/s/s

hint: use y = 0.5 a t² + Vot + yo    with y = 5, a = -32,  yo=6, Vo = 45
you need to "massage it" to get the from 0=  At² + B t + C.  solve for t/. (use the TI or the quadratic formula)


5) A basketball player passes the ball to a teammate who catches it 11 ft (y=11)  above the court, just above  the rim of the basket, and slam-dunks it through the hoop. (This play is an "alley-oop"). 
The first player releases the ball 5ft (yo=5) above the court with an initial vertical velocity of 21ft/sec/ (Vo=21)
How long is the ball in the air before being caught as it rises ?  

hint: use y = 0.5 a t² + Vot + yo    with y = 11, a = -32, yo=5, Vo = 21
you need to "massage it" to get the from 0=  At² + B t + C.  solve for t/. (use the TI or the quadratic formula)


6) The volcanic cylinder cone Puu Puai was formed in 1959 when a massive "lava fountain" erupted at Kilauea Iki Crater, shooting lava hundreds of feet into the air. When the eruption was most intense , the height y (in feet) of the lava t seconds after being ejected from the ground could be modeled by y = - 16t² + 350t.  (acceleration due to gravity in feet/s/s = -32)

A) What was the initial vertical velocity of the lava ?  (remember the equation of motion with yo = 0)
Vo = ________ ft/s

B)What was the lava's maximum height above the ground ?

hint: use V = at + Vo. with V= 0 , Vo =  given in A), a= -32

C) For how long was the lava in the air ? (solve y = 0 )



7) A  juggler throws a ball into the air, releasing it 5 feet above the ground (yo=5)  with an initial vertical velocity of 15ft/s.
(Vo = 15) She catches the ball with her other hand when the ball is 4 feet above the ground (y=4)
A) Find the equation of motion of the ball. y = __________________
(hint: use yo= 5, a = -32, Vo = 15)

B) Find how long the ball is in the air.
(hint: y = 4. solve for t)

8) REVIEW
QUIZ practice

9) On July 28, 1945 an airplane crashed in to the 78th and 79th floors of the empire state building in New York City.
Hundreds of pieces of debris fell 975 feet (yo=975) to the streets below. (Vo = 0, a= - 32ft/s/s)

A) how long did it take for the debris to reach the ground?

hint: use y= 0.5 a t² + Vot + yo  , solve for t if y = 0 (ground level). Vo =0

B) How much time after hearing the crash did the people on the street have to get out of the way of the falling debris ?
the speed of sound = 1000ft/s.

hint: find the time for the sound to cover 975ft. then subtract that from the time found in A)

10) Many birds drop shellfish onto rocks to break the shell and get the food inside. Crows along the west coast of Canada use this technique to eat whelks (a type of sea snail). Suppose a crow drops a whelk from a height of 20 ft (yo=20).

A) use an equation giving the whelk's height y  (in feet) after t seconds. (hint: a= -32ft/s/s, yo= 20, Vo = 0)
y= _________________________

B) Use your TI to find y when t = 0,0.1,0.2,0.3 ... , 1.4, 1.5
use the table t  y(t) to find the time it takes for the snail to hit the ground. (t that will cause y = 0)

hint: enter your equation into Y1 (Y= ). go to  TABLSET  and set TBLSTART = 0 and  TBL = 0.1,
the TI will automaticly generate y(t) for t = 1, 0.1, 0.2 .. got to TABLE to see these values.
Build a table. WATCH THIS MOVIE TO SEE WHAT TO DO  if you are lost.
Use this TABLE to find the time for the snail to fall. (find t for  y= 0)
UNDERSTAND that you are not interested in negative y. (does not make sense)

C) solve the quadratic function y = 0 to find the time of fall. Do you get the same value?

11)  On any planet, the distance  y (in feet) of a falling object t seconds after it is dropped can be modeled by:
d =  g/2 t²       (distance so don't worry about direction)

d is the distance covered by the object, g is the acceleration (in feet per second squared) due to the planet's gravity. For each planet in the table, find the time it takes for a rock dropped from a height of 200 feet to hit the ground.
It is easier if you first solve for t . t = ______ (the variable being g). Find t as a function of g.

hint: for any  planet  200 = 0.5 g t² or    400 = g t²     or    t = sqrt(400/g) = 20/sqrt( g)    sqrt means taking the square root.

planet	Earth	MArs	Jupiter	Neptune	Pluto
f (ft/sec2)	32	12	81	36	2.1
time (s)	___________	____________	_______________	_____________	____________

12) Niagara Falls in New york is 167 feet high (yo) . How long does it take for water to fall from the top to the bottom of Niagara Falls ? (a = - 32 ft/s/s, Vo =0, yo = 167)

13) You and your friend are playing tennis . You friend lobs the ball into the air, hitting it 3 feet above (yo) the court with an initial vertical velocity of 40feet per second. (Vo) . You back up and prepare to hit an overhead smash to win the point.

A) what is the model for height versus time  ? y(t) = y  = ____________  
hint: a = -32, Vo = 40, yo=3

B) At what time tmax does the ball reach its maximum height ymax above the ground ? What is the maximum height ymax?
hint: use V = at + Vo    with V = 0 at the top to solve for tmax
substitute tmax into y(t) to find ymax.

C) If you plan to hit the smash when the ball falls to a height of 8 feet (y=8)  above the court,
how long do you have to prepare for the shot ?

hint: set y=8  and solve the quadratic equation for t. You need first to get the form: 0=  At² + B t + C

D) What about if y = 6 feet ? and y = 9 feet ?

14) The height  of a tennis ball   hit by a racket can be modeled by the equation (with respect of time)
y = -16t² + 42t + 3
What do the coefficient of the quadratic equation represent ? (-16, 42 and 3  ) ?

LABS

1) reaction time. Material : ruler.
part1  part2

2) worksheet
par1   part2

3) free-fall device with sparks timers
part1  part2         part3  part4
part6    part7