His father was extremely disappointed, for he knew that mathematicians made no more money than musicians, but he finally agreed. 7
Galileo lectured at the University of Pisa in 1589 and was appointed to a chair in mathematics in 1592. He then moved on to the University of Padua, where he remained until 1610.
The Ballistics Problem
Within a few years Galileo made major contributions to the study of the trajectory of projectiles, which was of critical concern to gunners. It all started with his interest in gravity. Aristotle had said that all objects fall toward the earth with a speed that depends on their weight, and for years this appeared to be reasonable. It could easily be seen, for example, that very light feathers fell much slower than heavy stones. Galileo was skeptical, and according to legend he carried several balls of different weights to the top of the tower of Pisa and released them. The balls all struck the ground at the same time. Aristotle was wrong. Actually, there is no evidence that Galileo performed this experiment, but it is an interesting story nevertheless. 8
Galileo wanted to go further, however. It was now obvious to him that the balls accelerated as they fell, so they had different velocities at different positions above the earth. Indeed, the farther they fell, the greater their velocity, and Galileo wanted to measure their acceleration. But because objects fell so fast, it was difficult to set up a straightforward experiment. So he decided to slow things down. The best way to do this was to let the object roll down an incline. The object would speed up in the same way because gravity was still acting on it. Again, he noticed that the acceleration of the balls down the incline was independent of their mass. In other words, they all got to the bottom with the same speed, regardless of how much they weighed. A detailed study of this motion led to several important conclusions.
Earlier Galileo had discovered something similar with pendulums. While at church he had noticed objects at the end of long ropes swinging as a result of air currents in the church. Clocks were not available at this time so he used his pulse to time them, and he noticed that regardless of the distance they swung (called the amplitude) they took the same time to complete a swing. Again, itwas gravity that was pulling the weight downward (along with the current), causing the objects to swing. Galileo was never able to measure the acceleration of gravity, but we now know that it is 32 ft/sec 2 , and we know that it acts on all objects on earth. He did, however, show that the square of the period of the pendulum varied directly with the length of the pendulum.
Building on his discoveries in relation to gravity, Galileo decided to look at projectile motion carefully in an effort to thoroughly understand it. He imagined first of all that there was no air resistance, as he knew that the air around a projectile acted on it to change its motion. As a first step, it was best to ignore it. Second, he considered the forces that were acting on the projectile. Obviously the first force was the expanding gas from the gunpowder that thrust the projectile from the cannon barrel. Once it was out of the barrel, this force was gone, and the projectile would have a constant velocity if no other forces were acting on it. In stating this, Galileo was imagining a new concept that we now call inertia. All bodies in motion have a certain amount of inertia, and as a result of it they will continue in motion with the same velocity unless this inertia is overcome by an outside force. In the above case there was an outside force acting on the projectile after it left the barrel, namely gravity, and gravity would cause it to fall in the same way any object falls when released. The only difference in this case was that the projectile also had a horizontal velocity.
These results helped to give Galileo a better understanding of projectile motion. As a
Galileo lectured at the University of Pisa in 1589 and was appointed to a chair in mathematics in 1592. He then moved on to the University of Padua, where he remained until 1610.
The Ballistics Problem
Within a few years Galileo made major contributions to the study of the trajectory of projectiles, which was of critical concern to gunners. It all started with his interest in gravity. Aristotle had said that all objects fall toward the earth with a speed that depends on their weight, and for years this appeared to be reasonable. It could easily be seen, for example, that very light feathers fell much slower than heavy stones. Galileo was skeptical, and according to legend he carried several balls of different weights to the top of the tower of Pisa and released them. The balls all struck the ground at the same time. Aristotle was wrong. Actually, there is no evidence that Galileo performed this experiment, but it is an interesting story nevertheless. 8
Galileo wanted to go further, however. It was now obvious to him that the balls accelerated as they fell, so they had different velocities at different positions above the earth. Indeed, the farther they fell, the greater their velocity, and Galileo wanted to measure their acceleration. But because objects fell so fast, it was difficult to set up a straightforward experiment. So he decided to slow things down. The best way to do this was to let the object roll down an incline. The object would speed up in the same way because gravity was still acting on it. Again, he noticed that the acceleration of the balls down the incline was independent of their mass. In other words, they all got to the bottom with the same speed, regardless of how much they weighed. A detailed study of this motion led to several important conclusions.
Earlier Galileo had discovered something similar with pendulums. While at church he had noticed objects at the end of long ropes swinging as a result of air currents in the church. Clocks were not available at this time so he used his pulse to time them, and he noticed that regardless of the distance they swung (called the amplitude) they took the same time to complete a swing. Again, itwas gravity that was pulling the weight downward (along with the current), causing the objects to swing. Galileo was never able to measure the acceleration of gravity, but we now know that it is 32 ft/sec 2 , and we know that it acts on all objects on earth. He did, however, show that the square of the period of the pendulum varied directly with the length of the pendulum.
Building on his discoveries in relation to gravity, Galileo decided to look at projectile motion carefully in an effort to thoroughly understand it. He imagined first of all that there was no air resistance, as he knew that the air around a projectile acted on it to change its motion. As a first step, it was best to ignore it. Second, he considered the forces that were acting on the projectile. Obviously the first force was the expanding gas from the gunpowder that thrust the projectile from the cannon barrel. Once it was out of the barrel, this force was gone, and the projectile would have a constant velocity if no other forces were acting on it. In stating this, Galileo was imagining a new concept that we now call inertia. All bodies in motion have a certain amount of inertia, and as a result of it they will continue in motion with the same velocity unless this inertia is overcome by an outside force. In the above case there was an outside force acting on the projectile after it left the barrel, namely gravity, and gravity would cause it to fall in the same way any object falls when released. The only difference in this case was that the projectile also had a horizontal velocity.
These results helped to give Galileo a better understanding of projectile motion. As a
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