Goals: The student will learn that|
- Light and heavy objects fall at the same rate, as established experimentally by Galileo.
- Falling objects, and balls rolling down an incline, tend to accelerate at a constant rate a. Their velocity increases as v = at.
- The acceleration of free fall is g = 9.81 m/s2.
- With initial velocity u, v = u + at. Use this to calculate height attained by object thrown straight up.
- Distance covered is S = a(vinitial + vfinal)/2 = ut + gt2/2
- Fallacy of the "road runner paradox": when two different motions involve an object they act simultaneously--never consecutively. E.g. in shooting at a distant target, the gunsight make us aim at a higher point, so that the fall of the bullet brings it to the target.
- Possible addition by the teacher: artillery.
Terms: Acceleration, velocity (also initial, final and mean velocity), speed (=magnitude of velocity), "g" the acceleration due to gravity.
Stories and extras: Legend of Galileo dropping balls from Tower of Pisa (also of his timing the swing of a chandelier). "Road runner" cartoon, demonstration of feather falling on the airless Moon.
Hands on: Possibly, Galileo's experiment with the inclined plane
Use Galileo as starting point.
We remember Galileo for several things:
- He was the first to observe the heavens through a telescope (which he had designed and built)
- He was persecuted for advocating the Earth went around the Sun.
But he also:
- Was a pioneer of physics, introducing the idea of experiments--rather than try guess what nature might be doing, using logical
arguments, observe what it does do.
The legend is that as a boy, sitting in church with his family, Galileo became bored and to pass the time, he observed a swaying chandelier. Using his own pulse to time the swings, he discovered that they always took the same length of time, whether they were big or small.
The story may well be a legend--but Galileo did make such observations, and they led to a better design of clocks.
He certainly studied the fall of objects. Philosophers had argued that a heavy object fell faster. According to another legend, he climbed the leaning tower in the town of Pisa--a bell tower whose foundation settled soon after it was built, causing it to lean--and from the top dropped heavy and light balls. A helper on the ground observed that they arrived together.
Galileo learned to be careful in his experiments. It was known that thunder arrived some time after the lightning was seen.Firing a gun and timing the interval between the flash and the sound allowed the speed of sound to be measured. But did light travel instantly?
(Teacher may ask class--how would you find out?).
Galileo posted himself and a helper at night, a good distance apart, each with a lantern covered with a screen. Earlier, the helper was told--when you see the light, lift your screen. Galileo then lifted the screen and looked how long was the delay until he saw the return flash. The light, during that time, had to travel back and forth between him and his helper.
(Teacher may ask class--do you see any problem with this experiment?)
There was a delay, but Galileo realized it might just be the reaction time of the helper. He therefore repeated the experiment with the helper at a much greater distance. The delay was the same--and Galileo concluded that the velocity of light was too big to be measured this way.
Now back to falling objects. Galileo showed light and heavy ones fell together: he did not ask why; that question was later taken up by Newton.
[The question below takes matters too far afield. It should be skipped, unless (perhaps) a student raises it.
Sometimes people argue the reason is clear--take the big stone, divide it into 10 little stones. When those stones were together they fell the way the big stone fell, so when they are apart, shouldn't they fall at the same speed?
This leads to something important, which will however only be hinted at here (don't try to remember it at this stage). The basic answer: in air, yes, in water, no. Water slows down the fall of small stones much more effectively. But apart from air resistance, even completely free fall has something that opposes it. Fifty years later Newton finally understood how this happened, when he formulated his laws of motion. He named to opposing factor "inertia" and we now call it "mass." Ten small stones have the same inertia as the big one, so they fall just as fast.
The thing we note about falling objects is their velocity: it starts slow and gets faster and faster.
What is velocity, anyway?... Then go on to acceleration.
Guiding questions and additional tidbits
-- What is the velocity of a moving object?
The rate at which it covers distance.
-- What is speed?
The magnitude of the velocity. We may use the terms interchangably now, but in a later stage we will also have to pay attention to the direction of the velocity.
[Using "velocity" at this stage justifies the use of "v" and also of using "+" or "-" signs.]
-- If an object covers distance S meters in time t seconds--what is its velocity [or speed] v?
v = S/t meters per second, abbreviated m/s. (Strictly speaking, that is its average velocity.)
This, by the way is a formula, not an equation. The difference between equations and formulas is explained in the section on formulas.
-- If an object covers distance S miles in time of t hours, what is its velocity?
v = S/t miles per hour or mph (again, average velocity).
-- A biker pedals at 10 mph. What is the biker's speed in meters/sec?
1 mile=1609 meters.
16090 meters in 3600 seconds, S/t = 4.47 m/s
--What do you know about the speed of a freely falling object?
It constantly increases, at a steady rate.
-- What is the acceleration of an object?
The rate at which its speed grows.
-- Can acceleration be negative?
Yes. You jump from a table: the moment you leave it your speed increases and your acceleration is positive. When you hit the ground, your acceleration is negative.
[More generally, acceleration like velocity can have any direction. For now we do not go that far.]
-- What can you say about the acceleration of a freely falling object?
It is constant, equal to about g = 9.81 m/sec2.
-- How fast does a falling object move after t seconds?
-- How much is that in miles per hour?
First, the conversion factor needs to be derived.
Using a previous answer:
10 mph = 4.47 m/s
Divide both sides by 4.47
1 m/s. = 10/4,47 = 2.237 mph
v = (9.81* 2.237) t mph = 21.95 t mph.
After 3 seconds, more than 60 mph!
-- How is this modified if we throw the object downwards, and start it not from rest but with an initial velocity u?
Then u is positive, and adding it to the fall velocity gives v = u + gt
-- What if we throw it upwards?
Then u is negative, the opposite sign of g. If we want to only use positive quantities, define the positive speed u' = -u, so v = -u' + gt .
As long as v is negative, the object moves up, when it turns positive, it moves down.
-- After how long does it reach greatest height?
--How do we calculate the distance covered?
We multiply time t by the mean velocity.
-- What do we take as mean velocity?
Half the sum of initial and final velocities.
[Note that we have only guessed that with this definition, vmean*t=S. We have not proved it. Actually, this only works if the acceleration is constant. If the acceleration changes as time goes on, using this mean velocity in vmean*t usually does not give the right S. ]
--What then is the distance S?
S = 1/2[u + (u +gt)]*t = ut + (1/2)gt2
-- We throw a stone upwards with velocity u'. How high does it get?
The "upwards" direction was chosen at "negative" (opposed to the acceleration of gravity), so from the previous example, u = –u', and the time t needed for reaching the greatest height is u'/g. So:
S = ut + (1/2)gt2 = –u'(u'/g) + (1/2)g(u'/g)2 = –(1/2)(u')2/g.
The minus sign means, it is in the direction we chose as negative, above the starting point.
-- What is the distance with constant acceleration a?
S = 1/2[u + (u +at)]*t = ut + (1/2)at2
-- If a ball rolls down an inclined slope, because of its weight, does it accelerate?
Yes. If we neglect friction, the acceleration a is constant.
-- How did Galileo confirm the constant acceleration of a ball rolling down a slope?
The answer is in the "Stargazers" section.
-- When we throw a stone horizontally with a velocity w, how does it move?
Neglecting air resistance, it simultaneously moves horizontally with a velocity w while falling vertically with acceleration g.
If we plot the motion in (x,y), with x growing horizontally and y growing downwards, then
-- When "Road Runner" runs over the edge of a cliff, how does he move?
The same way. His horizontal motion continues at a steady rate, while at the same time he also gains a gradual downwards velocity. The result is a curve which gets steeper and steeper.
The way movie cartoons show it--first a horizontal motion (which suddenly stops), then a fall straight down, actually never happens!
-- When we fire a rifle towards a target at a rifle range, how does the bullet move?
The same way--mainly horizontally, but gradually also acquiring a downward fall.
-- When we fire a rifle towards a target in a rifle range, what do we do to score a bull's-eye?
We aim towards a higher point, so that the fall of the bullet brings it to the level of the bull's eye. The gunsight can be adjusted to raise the aim by an amount appropriate to the distance of the target.
-- When shooting arrows using a bow, how do we aim for the target?
Same way--aim higher. Here the shift is greater, because arrows travel more slowly. With a given distance, they have a longer time to fall and therefore drop a longer way down.
Optional: Artillery. A cannon gives the shell both an upward velocity u' and a horizontal one w. Three kinds of cannon exist:
(1) High speed cannon, like those on tanks. They fire near-horizontally, like rifles, and are aimed towards the target--with a small correction for the drop of the shell. However--to get this high speed takes a heavy, expensive cannon, with a strong recoil that somehow has to be absorbed.
(2) Howitzers, which are aimed around 45 degrees up, at which angle they get the greatest range. For the same range, therefore, they need less velocity, and one can use a lighter cannon. But to aim it, a forward observer is needed--presumably, linked by radio, telling the gun crew how to correct its aim.
(3) Mortars, firing at a steep angle, e.g. 60 degrees. That way the ground absorbs the recoil, allowing a light, portable gun (the smaller kinds can even be carried by hand), still firing a fairly heavy shell. However, it will not shoot very far--nor will it be very accurate, because the slow shell with a high path spends a long time in the air and is easily deflected by wind.