Galileo's Law of Inertia

This all ties in with the findings of Galileo and his law of inertia. Basically, we can restate the law of inertia as: Unless an object is acted on by a net, outside force, it will continue to move at the same velocity. Thus, Galileo recognized that the reason that objects on Earth always slow down and come to a stop after gaining an original velocity is because there is some force, which we call friction, constantly acting on the object in order to slow it down. Remove the friction and the object would continue to move at the same velocity forever.

Can we think of any examples of the law of inertia? One example is if we were to have two people pushing on an object with the same force from opposite directions, then the two forces cancel out. In this case, there is no net force, so there is no change in the velocity of the object. As another example, what would happen if we were to put some people in the bed of a truck and have them push on it? Obviously, we would find that the truck would stay still because the truck is pushing back on the people with the same force. Thus, there is no net outside force acting on the truck and its velocity doesn't change. What about examples where there is a change in the velocity? We have all experienced riding in a car when it brakes suddenly. If we are not braced against something, we are pitched forward. This is because originally we and the car are moving at the same velocity. When the car breaks, its velocity is lowered, but our inertia causes our velocity to remain the same. Thus, from our perspective, we move forward in the car until the seat belt catches us and decelerates us to the same speed as the car.

If all of this was discovered by Galileo in his law of inertia, why do we need Newton? Even though Galileo stated the law of inertia, he did not come up with a direct way of quantifying it, nor was he able to tackle the other major problem that we have talking about motion. Recall that we defined the average velocity as the distance traveled divided by the time it takes to go that distance. However, consider the following example. Lets sit in a car without moving for an hour, then very rapidly accelerate up to 75 miles per hour, travel at this speed for an hour, decelerate rapidly back to rest, and then sit for another hour. In this case, our average velocity would be 75 miles divided 3 hours, or 25 miles per hour. If we were to ask what the velocity was after 90 minutes, we would get two different answers. The true velocity was 75 miles per hour, but the average velocity is still 25 miles per hour. In order to get a better fit from the average velocity for the true velocity, we could try averaging over a shorter time period, keeping the interval centered on 90 minutes. In fact, the smaller we make the time interval, the better the approximation would be to the true velocity. Ideally, if we could let the time interval go to zero we should get that the average velocity becomes equal to the true velocity. The problem is, if we let the time interval go to zero, then we are dividing by zero, which is not defined mathematically. It was this problem that stopped Galileo from being able to completely describe the motion of objects. This was also the problem that Newton solved. 

From the lecture notes of  Dr. Daniel Suson, associate professor of physics at Texas A & M University, Kingsville. Notes found at http://newton.tamuk.edu/~suson/html/1411/physics.html#Galilean

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