Galileo's experiments (either thought or real) led to the following prediction:- Suppose two objects (a 1kg and 2kg mass) are dropped simultaneously from the same height above the earth in a vacuum, then the time for each to reach the earth would be the same.
1. I say that this prediction is only partially true.
2. Suppose the same experiment was repeated with each mass being released separately.
I say that the 2kg mass would collide with the earth in less time than the 1kg mass.
3. Suppose the same experiment was repeated with the 2kg and 1kg masses being released simultaneously but on opposite sides of the earth. "Obviously" the 2kg mass will collide with the earth before the 1kg mass.
The explanations are as follows:-
Let the mass of the earth be M and assume that the earth generates a gravitational field of strength g.
Suppose we drop an object of mass m from a height of 1m above the earth in a sealed vacuum. Let the gravitational field strength of the object be g'.
The force of the earth on the object will be F(e) = mg.
The force of the object on the earth will be F(o) = Mg'.
According to Newton's third law of motion F(e) = F(o) and hence mg = Mg'
The displacement of both the earth and the object may be calculated using
s = ut + 1/2 x a x t x t
Let the earth move X metres toward the object. Hence, in time t (time for collision) the object will move 1 - X metres.
For the earth:- X = 0 x t + 1/2 x g' x t x t
For the object:- 1 - X = 0 x t + 1/2 x g x t x t
Eliminating X between the two equations (just add them together) we get :-
t x t = 2 / (g + g')
Hence, by taking the square root of both sides we can find t (the time taken for the object and earth to collide). Notice that the time to collide is a function of g and g'.
PREDICTION NUMBER 1
If we increase the mass of the object to 2m its field strength will increase to 2g'. Gravity is a vector quantity. Hence, when we drop masses m and 2m one metre above the earth the total field strength, generated by the two masses, is 3g' (both fields are acting in the same direction). The net field acting on the earth is 3g'. Hence, the earth falls towards both masses at the same rate. The earth's field acting on both objects is g. Hence, both masses fall towards the earth at the same rate. The time to collide is the same for each mass and is given by
t x t = 2 / (g + 3g')
PREDICTION NUMBER 2
Dropping mass m by itself will result in a time of collision equation of:- t x t = 2 / (g + g').
Dropping mass 2m by itself will result in a time of collision equation of:- t x t = 2 / (g +2g').
Hence, the 2kg mass hits the ground in less time than the 1kg mass. The earth falls at a faster rate toward the 2kg mass compared to the 1kg mass because of the greater field strength generated by the 2kg mass.
It might be argued that for small masses the time difference is not measurable. This, I say, is irrelevant.
PREDICTION NUMBER 3
Suppose masses m and 2m are dropped simultaneously, in a vacuum, from a height of 1 metre on opposite sides of the earth. Mass m generates a field of strength g' and mass 2m generates a field of 2g'. Hence, the earth will fall faster toward the 2kg mass resulting in a lesser collision time.
SUMMARY
It should now be quite evident that the only instance when two objects of differing masses, released simultaneously 1 metre above the ground in a vacuum, will hit the ground at the same time is if they are in close proximity to each other.
Anyone who views force as a push or pull might have problems following the above. I suggest viewing my Blog regarding the concept of Force.
Tuesday, 1 September 2015
Thursday, 7 August 2014
The Australian Curriculum
The Australian Curriculum appears to have the same major flaw as State Curricula.
Some years ago I was teaching in a New South Wales Government School which had a significant aboriginal population. I was given the "bottom" year eight class and directed to teach them mathematics. Accordingly the Head Teacher gave me a copy of the common year eight syllabus and program.
My first lesson was centred around diagnostic testing. There were seven students in the class with three support persons present. For the next ten weeks I, the seven students, and three support persons made great progress which resulted in all students increasing their mathematical abilities and more importantly broadening their social skills.
With the end of term one now upon us the Head Teacher asked me for my program register. I stated that I had not quite finished writing my program and that I would have it written and signed off by the beginning of term two. He reminded me that he had given me a copy of the year eight common program at the beginning of the term. I told him that the material he had given me was unsuitable and was not teachable to those students. I also pointed out that it is impossible to teach the unteachable.
He was not happy! For the remainder of the year my class continued to make very good progress. The Head Teacher did not speak with me again regarding that class. In fact, he hardly spoke to me for the rest of the year.
The following year another teacher took the "bottom" year eight class. I later found out that he also threw out the common curriculum and taught a more suitable course of study. Apparently he and his class had a very fruitful year.
Having entire year groupings in mathematics and science follow the same curriculum in each of those subjects is fundamentally floored.
Some years ago I was teaching in a New South Wales Government School which had a significant aboriginal population. I was given the "bottom" year eight class and directed to teach them mathematics. Accordingly the Head Teacher gave me a copy of the common year eight syllabus and program.
My first lesson was centred around diagnostic testing. There were seven students in the class with three support persons present. For the next ten weeks I, the seven students, and three support persons made great progress which resulted in all students increasing their mathematical abilities and more importantly broadening their social skills.
With the end of term one now upon us the Head Teacher asked me for my program register. I stated that I had not quite finished writing my program and that I would have it written and signed off by the beginning of term two. He reminded me that he had given me a copy of the year eight common program at the beginning of the term. I told him that the material he had given me was unsuitable and was not teachable to those students. I also pointed out that it is impossible to teach the unteachable.
He was not happy! For the remainder of the year my class continued to make very good progress. The Head Teacher did not speak with me again regarding that class. In fact, he hardly spoke to me for the rest of the year.
The following year another teacher took the "bottom" year eight class. I later found out that he also threw out the common curriculum and taught a more suitable course of study. Apparently he and his class had a very fruitful year.
Having entire year groupings in mathematics and science follow the same curriculum in each of those subjects is fundamentally floored.
Thursday, 22 May 2014
DRUGS IN SCHOOLS
It is commonly thought that the major drug problems in some schools are related to substances such as "ice", cocaine, marijuana, etc or alcohol.
This is far from the truth. The biggest single drug problem in virtually all primary and secondary schools is smoking.
There are numerous reasons as to why students start smoking. It is my experience that the majority of students over 15 years of age want to give up their addiction.
In the past I have, "unofficially", helped students to give up their addictive habit. I point out to them that as I see it there are only three reasons why they continue to smoke. One they are stupid, or two the are addicted, or three they are both stupid and addicted. None of the students which I have helped to quit were stupid.
I would suggest that most schools do not want to admit that they have a smoking problem. It's an "image of the school" thing. I would also suggest that the so called educational programs related to smoking which are implemented in schools are generally a waste of time and money. They almost certainly do nothing for the many students who are addicted. It is completely understandable that the vast majority of addicted students cannot get through a school day without having a puff or two.
Many of the punishments which teachers dish out to students who are caught smoking are quite pathetic and do nothing to alter the students addiction.
If we want a society where a lot less people smoke then we need to properly address the problem of smoking in our schools.
This is far from the truth. The biggest single drug problem in virtually all primary and secondary schools is smoking.
There are numerous reasons as to why students start smoking. It is my experience that the majority of students over 15 years of age want to give up their addiction.
In the past I have, "unofficially", helped students to give up their addictive habit. I point out to them that as I see it there are only three reasons why they continue to smoke. One they are stupid, or two the are addicted, or three they are both stupid and addicted. None of the students which I have helped to quit were stupid.
I would suggest that most schools do not want to admit that they have a smoking problem. It's an "image of the school" thing. I would also suggest that the so called educational programs related to smoking which are implemented in schools are generally a waste of time and money. They almost certainly do nothing for the many students who are addicted. It is completely understandable that the vast majority of addicted students cannot get through a school day without having a puff or two.
Many of the punishments which teachers dish out to students who are caught smoking are quite pathetic and do nothing to alter the students addiction.
If we want a society where a lot less people smoke then we need to properly address the problem of smoking in our schools.
Images and Institutions
It is important for institutions such as churches and schools to maintain a positive image. Hence, leaders of these organisations are hand picked for this purpose.
I remember attending a staff meeting where a child protection issue was raised. A member of staff strongly suggested that every thing be kept "in house". That staff member was later promoted to the position of Deputy Principal and then Principal.
I suggest that this sort of action sends a clear message to staff members in both churches and schools.
I remember attending a staff meeting where a child protection issue was raised. A member of staff strongly suggested that every thing be kept "in house". That staff member was later promoted to the position of Deputy Principal and then Principal.
I suggest that this sort of action sends a clear message to staff members in both churches and schools.
Wednesday, 21 May 2014
VIOLENCE IN SCHOOLS
It seems that violence in schools throughout Australia is on the increase. Most Principals, and their executive staff, are committed to maintaining a positive image for their school. Hence, it is not too difficult to imagine that some classroom teachers and some students who have been subjected to violence are "sacrificed" for the sake of maintaining school image.
The Federal Government publishes certain statistics regarding schools. I always suggest to parents that the first set of data to view is school attendance rates. I further suggest to the same parents that if attendance rates are below 95% then it is quite possible that there is a significant problem with violence at the related school.
It is my experience that school attendance rates are over estimated and hence absenteeism is underestimated.
The Federal Government publishes certain statistics regarding schools. I always suggest to parents that the first set of data to view is school attendance rates. I further suggest to the same parents that if attendance rates are below 95% then it is quite possible that there is a significant problem with violence at the related school.
It is my experience that school attendance rates are over estimated and hence absenteeism is underestimated.
Monday, 3 February 2014
Zero
Numerous teachers of mathematics will tell you that x/x = 1, (x+1)/(x+1) = 1, (x-7)/(x-7) = 1, etc.
This is not quite correct. If x = 0, then x/x does not equal one. There is no inverse for zero and hence 0/0 has no answer.
In the second example above if x = -1 we again have 0/0. In the third example if
x = +7 we again have 0/0.
Hence, (x +9)/(x+9) = 1, but x cannot equal -9 is the complete answer.
Suppose you graph y = (sine x)/x. When x = 0 we have y= (sine x)/x = 0/0. Hence at x = 0, y = 0 on the number plane we put in a discontinuous point.
This is not quite correct. If x = 0, then x/x does not equal one. There is no inverse for zero and hence 0/0 has no answer.
In the second example above if x = -1 we again have 0/0. In the third example if
x = +7 we again have 0/0.
Hence, (x +9)/(x+9) = 1, but x cannot equal -9 is the complete answer.
Suppose you graph y = (sine x)/x. When x = 0 we have y= (sine x)/x = 0/0. Hence at x = 0, y = 0 on the number plane we put in a discontinuous point.
Sunday, 2 February 2014
RECYCLED WATER
Some communities in Australia refuse to drink recycled water. Some people say that they will never drink recycled water.
About 2000 years ago a Roman soldier dies in a secluded part of the North African desert. His body slowly decomposes and his water molecules evaporate into the atmosphere. From there his molecules become part of the world's water cycle. After 2000 years his molecules could be in the Indian Ocean, the Pacific Ocean, your local lake, your local water supply, etc.
One day you decide to drink a litre of water from your kitchen tap (this tap could be "anywhere" in the world). What is the most probable minimum number of water molecules that you will drink which were once in this soldiers body?
Most of my students usual guess between zero and one hundred. Onto the whiteboard I put up 0, 10,100,1000,...................1,000,000,000,000. I put the numbers up in powers of ten with the last number being ten to the power of 12. My students are asked to guess the closest number to the estimated answer.
The most probable minimum number is about 20,000,000 (20M). Hence, ten to the power of seven is the closest number which they can guess.
We ALL drink recycled water. The next time you drink a can of coke (which is mainly water) try estimating the minimum number of molecules which came from that Roman soldiers body!
About 2000 years ago a Roman soldier dies in a secluded part of the North African desert. His body slowly decomposes and his water molecules evaporate into the atmosphere. From there his molecules become part of the world's water cycle. After 2000 years his molecules could be in the Indian Ocean, the Pacific Ocean, your local lake, your local water supply, etc.
One day you decide to drink a litre of water from your kitchen tap (this tap could be "anywhere" in the world). What is the most probable minimum number of water molecules that you will drink which were once in this soldiers body?
Most of my students usual guess between zero and one hundred. Onto the whiteboard I put up 0, 10,100,1000,...................1,000,000,000,000. I put the numbers up in powers of ten with the last number being ten to the power of 12. My students are asked to guess the closest number to the estimated answer.
The most probable minimum number is about 20,000,000 (20M). Hence, ten to the power of seven is the closest number which they can guess.
We ALL drink recycled water. The next time you drink a can of coke (which is mainly water) try estimating the minimum number of molecules which came from that Roman soldiers body!
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