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Uniform Motion

When we graph the motion of a car or a bike and the resulting graph is a straight line we can find the slope of this line. The slope will be the same anywhere on this line and can be found by dividing the rise and the run. This led us to our operational definition for velocity (or speed) v = d/t. since the slope represents the speed the car or bike will have gone at a constant speed. We call this type of motion “uniform motion” – motion with a speed that is not changing. Most toy cars (those not remote controlled) go at a constant speed. Running is most of the time non-uniform (with changing speed). Can you think of examples when a runner is running at a constant speed? Real cars rarely go at a constant speed, only when cruise control is applied do we move uniformly.

For instance if you were to go to Boston, which is 120 miles away and you leave at 3:00 PM and get there at 6:00 PM will you have gone at a constant speed form door to door? Maybe on the Masspike, but certainly not when you start at the beginning and stop at the end. Do you know your speed at 4:30 PM when you were on the Masspike? You might want to calculate your speed using the numbers provided. It took you 3 hours to cover 120 miles. Given the definition for velocity you can find the speed as 120 miles/3 hour = 40 miles/hour. Now, is this your speed at 4:30? If you know the area you might figure out that halfway to Boston you would be on the Masspike. Is it likely that you will be going 40 miles/hour? Possible, yes, but not very likely. You will most likely be going 65 miles/hour. Is the calculation wrong? Certainly not. Then how come we didn’t get a result that makes sense? The answer is that we used the equation for uniform motion to find the speed of a trip, which is non-uniform. This is a good example for a mistake that is often made. You used an equation that calculated the speed, you did the math correct but the equation you used is not applicable to the situation. Is it in fact possible to find the speed at 4:30, provided you are not sitting in the car? You might guess that it is not. You could be stop at a rest stop, you could be speeding or you could be stuck behind a slow truck right at this moment. There is no way of knowing.

Then what did we calculate when we used the equation? What is represented by the speed of 40 miles/hour? It is called the average speed. But before we go any further we will have to define the average speed. The equation v = d/t calculates the average speed if t is fairly large (bigger than a few seconds for toy cars and a few minutes to hours for real cars, for instance).

Imagine one car going from Suffield to Boston with the speed of 40 miles/hour (somehow the car can get out of the garage at this speed) while you are going with your parents car (assuming you have a license) at the regular speed limits. If you both leave at 3:00 PM the first car, which is going at a constant speed of 40 miles/hour, will get to Boston at 6:00 PM sharp. You will make sure that you get to Boston at 6:00 PM as well, but at a speed that is non-uniform, that is it varies. You both start at 3:00 and get there at 6:00. Therefore your average speed will also be 40 miles/hour which is the same as the uniform speed of the first car. And this is the definition of average speed: The average velocity of a moving object is the uniform velocity required to start and finish the race at the same time as the object. This might sound a little confusing but read through it again keeping the example in mind that you just worked through.

Vectors

Velocity is one of those special quantities in physics. In physics quantities might not only have a unit, sometimes they even have a direction. Such quantities are called vectors. Vectors are specified by an arrow, whose length signifies the magnitude (or how big, or how strong). The arrowhead gives the direction. You are already familiar with some quantities for which it is important to specify the direction. Wind would be the most obvious one. It is not only important how strong the wind blows but also from which direction. The same is true for water currents.

But these are not all of them. Speed is another vector quantity. Going 50 mph can get you a lot of different places, only when specified 50 mph South, will you get to Hartford. Speed with a direction is called velocity. Acceleration is also a vector quantity. Mass, time and temperature are not.

 

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Vectors are added and subtracted (as well as multiplied and divided) differently then real numbers. Imagine you are in a boat going 4 m/s. The river is going at a speed of 3 m/s. What will your total speed be? You cannot answer this question without knowing whether you are going with or against the flow of the river. The answer could be either be 7 m/s or 1 m/s.

What if you are trying to cross the river. Now the answer is neither 7 m/s nor 1 m/s, but something in between. Imagine you are crossing a river that flows from left to right. You are going up on the screen. Remember that vectors can be represented like arrows. Take a piece of paper, use a ruler and put the first vector up. Make it four units long (either cm or inches). This is your velocity in the boat. AT the end of this arrow put another arrow pointing right. Make this vector 3 units long. You should now have the beginnings of a right-angled triangle. Your total velocity is the diagonal going from the start of the first vector to the head of the second. You can find the length by either measuring it or using the Pythagorean Theorem. Or, you might notice that this is a 3-4-5 triangle, therefore the diagonal is 5 units long. Your velocity is thu8s 5 m/s up and to the right. You will find more example in your workbook.

 

 

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