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1. Some Definitions
- Speed
Speed describes how fast something is moving. A simple example would be to look at your car's speedometer while you are driving. This tells you the speed at which you are traveling. Notice that when you look your car's speedometer, it only tells you the speed at which you are traveling. It does not tell you the direction in which you are traveling. Of course, this seems obvious to anyone who drives a car, but I just wanted to make the point that speed does not involve a direction.
How does speed describe motion? Well, this is also fairly obvious. For instance, when a car is moving at safe and legal speed of 55 mph (miles per hour), it will travel a distance of 55 miles in one hour of time. If it were moving at a speed of 100 mph, it would travel a distance of 100 miles in one hour of time. No surprises here.
Next, let me define something called average speed. Average speed is defined a follows.
Average Speed = [Distance Traveled]/[Time Taken to Travel that Distance]
For instance, let's say that it takes you 2 hours to travel a distance of 100 miles. Using the above formula, your average speed during those 2 hours would be 50 miles/hour or 50 mph. This is because 100 miles divided by 2 hours is 50 miles/hour.
Next, let's contrast average speed to instantaneous speed. Well, as you might have guessed, instantaneous speed is the speed at which you are currently traveling at the moment. For instance, if you are driving along and look down at the speedometer, your instantaneous speed at that moment would be what was displayed on your speedometer.
So, how does instantaneous speed differ from average speed? Well, let's go back to the example above. One way to get an average speed of 50 mph over 2 hours would be to simply drive at 50 mph all the time. In this case, the average speed would be the same as the instantaneous speed. However, let's say your foot is not the steadiest part of your body. If this is true, then your instantaneous speed would fluctuate a lot. However, if you still manage to cover 100 miles in 2 hours, even though your speed was fluctuating, then your average speed would still be 50 mph, but your instantaneous speed during those two hours of driving would not always be 50 mph.
Let me give a concrete example to illustrate this point. Let's say you drive at 25 mph for the first hour and then you drive at 75 mph for the second hour. In this case, your instantaneous speed during the first hour would be 25 mph at any moment. Your instantaneous speed during the second hour would be 75 mph at any moment. However, if you average those speeds (add them up and divide by two), you will find that the average speed was 50 mph. I hope this serves to illustrate the difference between instantaneous and average speed.
- Velocity
Velocity is very similar to speed except that it involves a direction as well as the speed. To determine the velocity of an object, you would need to know the object's speed and direction. To measure velocity in a car, you would need a speedometer and a compass. In essence, velocity gives you more information about an object.
- Thought Question: Does constant speed necessarily imply constant velocity?
Think about this before reading the next paragraph. In particular, think about the definitions of speed and velocity. Focus on the differences between the two definitions.
The answer is no because the direction could be changing. Recall that velocity involves both speed and direction. Therefore, for a velocity to be constant, the speed and direction would both have to be constant. So, what would an object moving at a constant velocity look like? One answer would be that the object would not be moving at all. This is fine because the speed is 0 mph and the direction is not changing. The only other possible answer would be that the object is moving at a constant speed in a straight line. The constant speed part shouldn't be confusing. But, why does the object have to move in a straight line? Well, if it didn't move in a straight line, then the direction would be changing, and, as a result, the velocity would be changing and not constant.
To summarize, the only way an object can have a constant velocity would be if it was sitting still or if it was moving in a straight line at a constant speed. There are no other alternatives.
- Acceleration
Acceleration is defined as the change in velocity over time. This is also one of the concepts that people new to physics have a trouble grasping initially. Any time an object's velocity is changing, we say that the object is accelerating. This brings up an important point. In common language, when things speed up, we say that they are "accelerating", and, when they slow down, we say that they are "decelerating". However, in the language of physics, we say that both objects are accelerating, not because both objects are speeding up, but because both objects have changing velocities. This can be a confusing point at first. When I am using the word "accelerating" in terms of the common definition of the word, I will put it in quotes. For the physics definition, I will not use quotation marks.
Finally, there is one more warning I'd like to offer about the definition of acceleration. Since acceleration involves a change in velocity, an object might be accelerating even though its speed is constant. Why is this possible? Well, it goes back to the difference between speed and velocity. Remember that velocity involves both speed and direction. So, a changing velocity does not have to necessarily involve a change in speed. It could just involve a change in direction.
For example, consider a car moving at a constant speed of 55mph while turning in a circle. The car's velocity is not constant, even though the speed is constant. This is because the direction of motion is constantly changing while the car is turning. Since the direction is changing, even though the speed is not, the velocity is changing. (Remember, the velocity involves both speed and direction.) As a result, the car is accelerating, even though it is neither speeding up or slowing down. The car is accelerating because its velocity is changing.
Finally, before moving on to the next section, think about what causes an object to accelerate?
- First Law of Motion
An object that is at rest will remain at rest unless a nonzero net (or total) force is exerted on it. This one is fairly easy to believe and sounds intuitive enough. However, the next part of the first law might sound less plausible. Simply stated, an object moving at a constant velocity will continue to move at a constant velocity (moving at a constant speed and in a straight line) unless a nonzero net (or total) force acts upon the object. Recall that constant velocity means that the object is moving at a constant speed and in a constant direction.
At this point, you might be thinking to yourself about something you saw in the world that contradicts the statement I just made. For instance, think of a car rolling in a straight line while in neutral. If what I stated above were true, then the car should be able to roll in a straight line at a constant speed forever. This is obviously not true in real life because everyone knows the car eventually comes to a stop. This certainly seems to prove that Newton's first law of motion is false. Or does it? I assert that the above observation is consistent with Newton's first law which states that an object moving at a constant velocity will continue moving in that fashion unless a nonzero net force acts upon that object. You might already know the answer.
It is true that the car would continue to move in a straight line at a constant speed if there was no net force acting on the car. However, is there really no net force acting on the car? In fact, there is a nonzero net force acting on the car, causing it to slow down. And, you probably already know what that force is. The force responsible for slowing down the car is friction. Therefore, the above observation about a car slowing down while in neutral is not inconsistent with Newton's first law of motion. It just seems to contradict it at first. If this is confusing, pause for a moment and think about it for awhile.
This is a pretty surprising fact to most people when they first hear it. If we were able to remove all the friction between the ground and a ball, once you start the ball rolling, it would roll on forever in a straight line.
This is a perfect tie-in to Newton's second law. We just discovered that objects like to move in straight lines and at constant speeds unless a force acts upon them. In fact, when a force acts on an object, the force causes the object to change its velocity. In other words, forces cause objects to accelerate.
- Second Law of Motion
Simply stated, a force causes an object to accelerate. Whenever you see an object accelerating, there must be an external force acting on the object because, as stated in Newton's first law, objects move at a constant velocity unless acted upon by an outside force.
Mathematically, Newton's second law of motion can be expressed by the following formula: a = F/m where a = acceleration, F = force, and m = mass.
What this formula tells us is that force causes an object to accelerate. However, it also tells us that the acceleration an object feels, in response to an applied force, does not solely depend on the amount of force applied. It also depends on the mass or inertia of that object. It also tells us that the more mass an object has, the less it accelerates in response to an applied force. This makes intuitive sense. For instance, if I apply the same force to a cotton ball and an elephant, the cotton ball would experience a greater acceleration than the elephant because the elephant has much more mass or inertia. Therefore, an object with a greater mass has a better tendency to resist a change in its motion when an external force is applied to that object. In other words, we say that the elephant has more inertia than the cotton ball.
Part of the beauty of math is that all this can be elicited from looking at the formula above in a much more compact form without reading an entire paragraph of explanation.
- Third Law of Motion
Newton's third law states that whenever a force is exerted, an equal and opposite force arises in reaction to this force. In other words, every force has an equal and opposite reaction force.
For example, when you push on a wall, the wall will also push back on you with an equal and opposite force. By the way, the "Newtons" in the figure above is the unit in which force is measured. In what follows, I will write "N" in place of "Newtons". For example, 5 Newtons of force will be written as 5 N.
Some of you might be wondering why you don't move backwards even though the wall is pushing you backwards. How very astute. The reason why you don't move backwards when you push against a wall is because static friction is pushing you back with an equal amount of force to the right so that you don't move anywhere. In the above example, static friction would be pushing the person to the right with 5 N of force, so that the person would experience zero total (or net) force, hence the person does not move.
This brings up an important point, i.e., that forces add up. We will come back to this point later when we discuss force in more detail.
So, if Newton's third law is true, and the wall pushes back on us just as hard as we push back on it, there must be some way of seeing that in the real world. Well, there certainly is. If you have ever gone ice skating or in-line skating (notice I'm not using the word Rollerblading) or roller skating (if you are really old), you can probably recall the example to follow.
Recall that the only reason why you didn't move when you pushed against the wall was because there was friction pushing you back to the right. Well, if you go skating, ice skating for example, you are reducing the friction between you and the floor because ice is very slippery. As a result, there isn't enough friction to compensate for the wall pushing you back. If you push against the wall while ice skating, you will move backwards as a result of the reaction force to you pushing against the wall.
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