Free-Fall

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 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 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 g = -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 d. positive if the object is above the origin. negative if it is below.

Exercise:

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 represent ?
B) Use the data in the table to plot a position-time graph (meaning x=time and position = y)
What kind of curve is this ? _________ The position varies as the square of the ________
C) optional. For Math lovers:
Find the slope of the curve at the end of 2.0s and 4.0s on the
position-time graph. Do the values agree with the table of velocity ?
F) Does this curve agree with the equation d = 1/2 gt² ?
To find out , using the formula, find d when t = 3s. 


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:


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 trajectory : (don't forget the sign)

V = _____________ and a = ____________

2) 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/ (see above for explanations)
(with Vi = 20m/s, a = -10m/s/s)

time (s)	displacement (m)	velocity (m/s)
0
1
1.5
2
2.5
3
3.5
4
5
6

3) 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 ?

OTHER PLACES

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.

EXERCISES

4) A high-wire artist missteps and falls 9.2m to the ground. What her speed on landing ?
(use one of the big five. )

5) A cat falls out of a tree, dropping 16 m to the ground. How long is the cat in the air ?

6) 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

7) 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.

8) A ball is thrown vertically upward with an initial velocity of 98.5 ft/s
(a) how high the ball go,
(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 ?

9 is an option. For Math lovers
9) 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) Compute the distance traveled by each car for times of 1s, 2s, 3s, and 4s
hint: For car A use d_A = 0.5 a t² with a = 4.5 m/s/s and for car B use d_B = V t that is d = 10t
b) For each car plot, on the same graph, distance vs time.
c) 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.
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.

10) 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: look at the picture above.
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
d) How high above its starting point is the ball 2 seconds after it is released ?
hint: use d = 0.5 a t²

optional. for Math lovers.
11) Using a TI or excel stat. functions/
Members of a Physics class stood 25 m apart and used stopwatch to measure
the time a car driving down the highway passed ech person. The data they compile are shown:

time (s)	position (m)	time (s)	position(m)
0	0	5.9	125.0
1.3	25.0	7.0	150.0
2.7	50.0	8.6	175.0
3.6	75.0	10.3	200.0
5.1	100.0

Use a graphing calculator to fit a line to a position-time graph of the data and to plot this line.
Be sure to set the display range of the graph so that all data fit in. FInd the slope of the line.
WHat does the slope indicate ?

12) An astronaut drops a feather from 1.2m above the surface of the moon.
 If the acceleration of gravity on the moon is 1.62 m/s² 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)
QUIZ practice

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