Well, the cat is already out of the bag on this one.
As stated in Newton's 2nd Law, an applied force is what causes an object to accelerate. The force along with the mass of an object determines the degree of response (acceleration) that the object experiences as a result of the force. This is an important point that some people gloss over when they first learn physics.
And, it is precisely the reason why I have written the mathematical formulation of Newton's 2nd Law as a = F/m instead of the more familiar F = ma. Nothing has changed by rewriting the formula. It was done purely for the sake of clarity. By writing it as a = F/m, we are emphasizing that it is force which causes an object to accelerate and not the other way around. Acceleration does not cause a force.
(As an aside, it was Roger Freedman who suggested writing Newton's 2nd Law as a = F/m instead of the more familiar F = ma.)
- Mass
In the form, a = F/m, it is also easier to determine the effect of mass on the amount of acceleration that an object experiences as a result of an applied force. Since mass, m, appears on the bottom, we know that mass is inversely related to acceleration. What this means is that if the same amount of force is applied to two objects, the object with the larger mass will experience a smaller amount of acceleration (or response to the force). Remember the example on the previous page about pushing the cotton ball and the elephant with the same amount of force. The reason why this is apparent from the formula above is because we know that, if we divide by a larger number, the result will be a smaller quantity. Therefore, when you have a more massive object, you are, in effect, dividing by a larger number thus resulting in a smaller quantity, acceleration, in this case.
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- Forces Are Additive
Next, I want to come back to another point I made earlier, namely that forces add. For this discussion, I will just consider motion in one dimension. This fact is so obvious it almost needs no introduction, but, nevertheless, I'll provide an example to motivate the point.
If you and a friend push a box with the same amount of force but in opposite directions, the box won't move (not a very good thing to do when moving things). This is because the forces add up. In this case, the forces cancel one another. Let me provide another obvious example to further illustrate the point. You already know that it is easier to push a stalled car when two people are pushing it instead of just one person. Well, this is obvious you might say, but it provides an example of the
additive nature of forces. Let us assume that it takes 50 Newtons of force to move a stalled car. If you were just pushing it by yourself, you would have to provide all of this force by yourself. However, if two people are pushing the car, your combined total would have to be 50 Newtons. So, you could push 20 Newtons, and your friend could push 30 Newtons so that the total force would still be 50 Newtons. However, we know that this example is completely hypothetical because you would never let your friend push more Newtons than you.
- Direction
Now that we know there is a relationship between force and acceleration, let me introduce another point that might sound obvious. As you already know, acceleration also involves a direction because the definition of acceleration involves velocity which has a direction associated with it. Likewise, force has a direction, namely the direction in which it is applied. So, it should come as no surprise that the direction in which the force is applied is also the direction in which the object accelerates. This should be obvious because the acceleration that an object experiences is the direct response to the applied force. The object should accelerate in the direction in which the force is applied. If it didn't, then it would seem weird. Recall that acceleration is the change in velocity over time. Therefore, a force causes an object to change its velocity. Remember Newton's 2nd Law. Objects naturally like to keep a constant velocity (zero velocity is just a special case of constant velocity) unless they are perturbed by a nonzero net (or total) force.
- Acceleration in Detail
- Figuring Out the Direction of Acceleration.
Instead of figuring out the direction of acceleration. It is often more intuitive to figure out the direction of the force first. This is because most people have a better idea of force than they do of acceleration. Once we figure out the direction of the force, we automatically know the direction of the acceleration because they are both in the same direction. Recall above that acceleration is an object's response to a force that is applied to that object. Since the acceleration is the object's response to a force, the acceleration must be in the same direction as the force. Let me give a couple of examples to illustrate the point. Once again, we will just consider motion in one dimension.
- Example 1
Consider an object that is moving to the right and speeding up. Is the object accelerating? If so, in which direction is the object's acceleration?
Well, the first question is easily answered. All we have to do is recall the definition of acceleration. Anytime an object's velocity is changing, the object is also accelerating. Also, since velocity involves both direction and speed, if either of these is changing, then the velocity is changing. In this case, the speed is changing (increasing, in this case) while the direction is remaining constant, therefore the object's velocity is changing, and, hence, the object is accelerating.
Now that we've answered the first question, let us move on to the second question. To answer this question, let me pose another question. If the object is moving to the right and speeding up, is there a force acting on the object? If so, in which direction is that force? Well, since the object's velocity is changing (object is accelerating), there must be a force. This is because of Newton's 1st Law of Motion
According to the 1st law of motion, if an object is not moving at a constant velocity, then there must be a net force acting on that object. Now that we've established there is a net force acting on the object, let us determine the direction in which it is exerted. This answer is fairly obvious. If the answer isn't obvious, picture a car in neutral moving to the right. Which direction would you have to push the car to get it to speed up while moving to the right? The answer is that you would have to push to the right.
Since we now know that the direction of the force is to the right and that the direction of the force and acceleration must be in the same direction, we can conclude that the direction of the acceleration is to the right.
Before moving on to the second example, let me draw some useful conclusions from this example. In common language, this example involves an object that is "accelerating" instead of "decelerating". (Remember that when using the word "accelerating" to mean the common interpretation of speeding up, I will put it in quotes.)
The first conclusion that can be drawn is that, when the force is in the same direction as the velocity, the object will speed up in the same direction as the velocity. In other words, when the direction of the force is in the same direction as the velocity, the object "accelerates". In this example, the direction of the velocity is to the right because the object was
moving to the right. This is in the same direction as the force, which was concluded to be pointing to the right, as well.
The second conclusion that one can draw from this example is that an object speeds up in the direction of its velocity when the direction of its acceleration and its velocity are the same. Alternatively, one could say that, when the direction of acceleration is the same as the direction of an object's velocity, the object will "accelerate". This obviously follows from the first conclusion because we know that the direction of the force and acceleration are the same.
What I wanted to do in this example was to illustrate the difference between the common definition of "accelerate" (speed up) and the physics definition of accelerate. Therefore, when we say an object is "accelerating", what we really are saying is that the direction of the object's acceleration and velocity are the same. If all this is confusing, this is the reason that it is often easier to determine the direction of the force first and then to use it to find the direction of the acceleration.
- Example 2
Consider an object moving to the right and slowing down. Is the object accelerating? If so, in which direction is the object's acceleration?
Before reading further, see if you can figure this problem out by using what you learned in the previous example. Go through the same thought process above and you should be able to get the answer.
Once again, the first question is easily answered. Since the velocity is changing in this example, the object is accelerating (no quotations implied). Because the speed is changing (decreasing in this case), the velocity is changing, therefore the object must be accelerating.
To answer the second question, it is, once again, easier to think of the direction of the force. To help us do this, let us rephrase the questions above. If an object is moving to the right and slowing down, is there a net force present? If so, in which direction does this net force have to be in order to slow down the object while it is moving to the right?
Well, since the object's velocity is not constant, there must be a net force acting on that object, according to Newton's 1st Law of Motion. As for the second question, this should be obvious as well. Once again, consider a car moving to the right and in neutral. In which direction would the force have to act in order for it to slow down? The answer is that the force would have to be to the left. This basically solves the problem. If we know the direction of the force, then we know the direction of the acceleration, as well, because they are always in the same direction. Therefore, the direction of the acceleration points to the left.
Finally, as in the above example, we can draw some useful conclusions. In common language, this example involves an object which "decelerates" because it slows down. Therefore, the first conclusion is that when the net force and the velocity are in opposite directions, as is true in this example, the object slows down or "decelerates". Likewise, we can say that an object "decelerates" when the direction of its acceleration is opposite to that of its velocity. Therefore, when we say an object is "decelerating", what we
really are saying is that the direction of its acceleration is opposite to the direction of its velocity.
Hmmmm... Something seems fishy about that last conclusion. Perhaps a quote from the Pinky and the Brain show would be appropriate here. "Pinky, are you pondering what I'm pondering?" If the net force is opposite to the velocity, what happens to the object after its velocity drops to zero? A very good question, indeed.
Well, the answer isn't that hard and is probably apparent to you already. If the force remains in the same direction (left in the example 2 above) after the object's velocity drops to zero, the object will begin to speed up to the left. After the velocity drops to zero, the problem resembles that of example 1 where the force and the velocity are in the same direction.
Therefore, if a net force is applied in a direction opposite to the object's velocity, the object will initially continue moving in the same direction while slowing down to zero after which it will begin to speed up in the direction of the force, which is opposite to the direction in which the object was initially moving. In other words, when a net force is applied opposite to the velocity of an object, the object will initially "decelerate" to zero (while moving in the same direction that it was initially moving) and then it will begin to "accelerate" in the direction opposite to the initial direction that the object was moving in.
All that might seem confusing but give it some time and thought, and it should become clear. To think about it, all you have to do is follow the thought process that went into example 2 above until the object reaches zero velocity. After that, use the thought process that went into example 1 above.
There was a lot in this section, so let me just reiterate the two main conclusions.
- When the net force is in the same direction as the object's velocity, the object will continue to move in the same direction, while its speed will increase. In other words, the object will "accelerate".
- When the net force is opposite to the direction of the object's velocity, the object will initially continue to move in the same direction while slowing down. After the object reaches zero velocity, it will begin to speed up in the direction opposite to the direction of its initial velocity. In other words, the object will initially "decelerate" while moving in the same direction. Once the object "decelerates" to zero, it will then "accelerate" in the direction opposite to its initial velocity.
You might have wondered why I didn't consider the case of zero net force. In fact, we have already considered this possibility. This possibility is exactly what Newton's 1st Law of Motion describes. If an object is subjected to zero net force, then it will continue to move at a constant velocity, meaning that it will continue to move in a straight line at a constant speed. Furthermore, it will continue to move in this fashion until it is subject to a nonzero net force.
- Constant Acceleration
Constant acceleration is an important special case where we can learn about the motion of an object without doing a lot of actual math. You might be tempted to say, "Hold it there, Tex." We just spent an entire section talking about how useful it is to think of force when we want to determine the direction of the acceleration. You might have even thought to yourself, "Why even bother with acceleration? What good is it for?"
Well, the answer is that force can only tell us some general things about the motion of an object, like whether it will speed up or slow down, but it can't tell us the specifics, like by how much it will speed up or slow down. For that, we will need acceleration.
Recall that acceleration is defined as the change in velocity over the change in time. In other words, acceleration tells us how an object moves by telling us how its velocity changes over time.
- Thought Question:
Knowing the definition of acceleration, if an object is experiencing a constant (nonzero) acceleration, does that mean the object's velocity is also constant? Hint: Since I said it was a thought question, the answer might be slightly tricky.
Well, to answer this question, let us recall what it means for an object's velocity to be constant. If an object's velocity is constant, then the object's speed and direction are both constant. So, what does this imply about the acceleration of the object? Well, since acceleration is defined as the change in velocity over time, the only way an object's velocity can be constant is if the object is not experiencing any acceleration. In other words, the only way an object's velocity can be constant is if its acceleration is zero.
Back to our thought question. Since we want to consider the case of a constant nonzero acceleration, the acceleration is not zero. As a result, the velocity is not constant. The answer to the thought question is that a constant nonzero acceleration does not mean that the object's velocity is constant. In fact, as we've already discussed, the only way that an object's velocity can be constant is if the object is not accelerating.
An alternative way to think about this is to think about force and Newton's 1st Law. Recall that the only way an object's velocity can be constant is if there is no net force on the object. If the net force on an object is zero, then the object's acceleration is also zero. This follows from the formula: a = F/m, which comes from Newton's 2nd Law. Once again, reestablishing the fact that a constant velocity implies that the object's acceleration is zero.
So, we've established the conclusion that a constant nonzero acceleration does not mean that the object's velocity is constant. So, what does an object do when it is experiencing a constant nonzero acceleration? I won't keep you in suspense any longer. When an object undergoes a constant nonzero acceleration, the object's velocity changes by the same amount over some specified amount of time. In other words, its velocity constantly changes by the same amount over some set time interval.
- Example 1.
Let me give an example to illustrate the point. Let us say that a car is undergoing a constant acceleration where its velocity changes by 10 mph to the right after every 10 seconds. Note that I have specified a direction for the change in velocity because acceleration involves both an amount of change in speed and a direction in which it changes. For this example, let us assume the car is initially at rest (0 mph).
- Initially, the car is not moving at all.
- After 10 seconds, the car is moving to the right at 10 mph.
- After another 10 seconds, the car is moving to the right at 20 mph.
- After yet another 10 seconds, the car is moving to the right at 30 mph.
By now, you should be getting the picture. Constant acceleration means that the velocity changes by the same amount over some set time interval. In this example, the car picks up an additional 10 mph to the right for every 10 seconds it undergoes the constant acceleration.
- Example 2.
If a car is initially at rest, what is its velocity after 25 seconds if it undergoes a constant acceleration of 3 mph to the left over a 5 second interval?
You should be able to figure out this problem by using the same reasoning as in the example above. Try it first before looking at the solution below.
The answer is 15 mph to the left. It is very important that you also specify the direction because the question asked for velocity which involves both speed and direction. Here's how we arrived at the solution.
- Initially, the car is at rest.
- After 5 seconds, the car is moving at 3 mph to the left.
- 10 seconds after the car started, the car is moving at 6 mph to the left.
- 15 seconds after the car started, the car is moving at 9 mph to the left.
- 20 seconds after the car started, the car is moving at 12 mph to the left.
- 25 seconds after the car started, the car is moving at 15 mph to the left.
The main thing to see is that 3 mph is added to the car's speed for every 5 seconds the car undergoes the constant acceleration.
- Example 3.
In the two examples above, we started the car out with an initial velocity of zero. While that might have made things simple, there are going to be cases when the initial velocity won't be zero. So, let's try our hands at an example where the initial velocity is not zero. There shouldn't be any surprises here because we are just building on what we have just learned.
In this example, let us assume the car is already traveling to the right with a velocity of 55 mph. Let us also assume that the car then undergoes a constant acceleration of 5 mph/10 seconds to the right. This is just a fancy way of saying that the car is experiencing a constant acceleration where its velocity changes by 5 mph to the right for every 10 seconds. Notice, once again, that I have specifically stated a direction because acceleration requires a direction.
What is the velocity after 30 seconds?
- Initially, the car is moving at 55 mph to the right.
- After 10 seconds, the car is moving at 60 mph to the right.
- After 20 seconds, the car is moving at 65 mph to the right.
- After 30 seconds, the car is moving at 70 mph to the right.
The answer is that the car's velocity would be 70 mph to the right after 30 seconds.
Notice that a direction is also required because the question asked for velocity which requires a direction as well as the speed.
You should be seeing a trend here. When the car's velocity and its acceleration are in the same direction, the car's speed increases. This is in agreement with what we discussed above.
When the velocity and the acceleration are in the same direction, we add to the speed by the amount of change in speed caused by the acceleration over the specified amount of time. In this example, we added 5 mph to the right for every 10 seconds the car undergoes the constant acceleration, independent of what the velocity is.
Since we are doing so well, let's just ask a question on the side. What happens to the velocity of the car after 30 seconds, if the acceleration becomes zero after the car in the example above accelerates for 30 seconds?
Well, this isn't too hard of a question if you know what happens to a car when it is not feeling any acceleration. Recall that zero acceleration means that the net force is also zero. Once again, refer to Newton's First Law of Motion. for the answer.
After 30 seconds, the car's velocity is 70 mph to the right. At that point, if the acceleration becomes zero, the net force on that car is also zero. Therefore, according to Newton's First Law, the car's velocity will remain constant. The answer is therefore that, after 30 seconds, the car's velocity will remain at 70 mph to the right because it is no longer undergoing an acceleration.
- Example 4.
This will be the hardest example of the bunch because this example will involve an acceleration that is opposite to the direction of the initial velocity. However, don't panic because you already know everything that you need to solve this problem.
Assume that the car has an initial velocity of 15 mph to the right and that it experiences an acceleration to the left of 5 mph/4 seconds. What is the car's velocity after 20 seconds?
Before jumping into this problem, let us see if we can get an idea of what the car will do in general when the acceleration is opposite to the initial velocity. It will be helpful to think of force in this case. Remember that an acceleration is caused by a force and that the acceleration is in the same direction as the force. Therefore, when we say that the acceleration is opposite to the velocity, we are also saying that the force is in a direction that is opposite to the velocity.
Lucky for us, we have already discussed what happens in this case. Refer to this section above. When the force is opposite to the velocity, the car will first slow down to zero. At this point, the force and the velocity will be in the same direction, and the car will then speed up in the direction of the acceleration. Therefore, we know that in our example here, the car must first slow down while moving to the right and then it should speed up while moving to the left.
With that said, let's proceed to solve this example. In all three examples above, we added to the speed of the car because the velocity and the acceleration were in the same direction. In this example, we have a different case, namely that the velocity and acceleration are in opposite directions.
Can you guess what we need to do to the speed when the velocity and acceleration are in opposite directions?
Since we know that the car must slow down when the velocity and acceleration are in opposite directions, it would seem logical to suggest that we subtract from the speed. If this is what you thought, then you are absolutely correct.
When the velocity and acceleration are in opposite directions, we subtract from the speed by the amount of change in speed caused by the acceleration over the specified amount of time.
Let us now proceed to solve this problem. Recall that the car's initial velocity is 15 mph to the right and that it is undergoing a constant acceleration to the left of 5 mph/4 seconds.
- Initially, the car's velocity is 15 mph to the right.
- After 4 seconds, the car's velocity is 10 mph to the right.
(Note that we subtracted from the speed by 5 mph because the velocity and the acceleration are in opposite directions.)
- After 8 seconds, the car's velocity is 5 mph to the right.
(Once again, we subtracted because during this time, the car's velocity and acceleration were in opposite directions.)
- After 12 seconds, the car's velocity is 0 mph.
(This is for the same reason as above because, during the time between 8 seconds and 12 seconds, the acceleration and velocity were in opposite directions.)
- After 16 seconds, the car's velocity is 5 mph to the left.
(This is because, after the car's velocity reaches zero, the acceleration and velocity are in the same direction. Therefore, we start to add to the speed in the direction of the acceleration.)
- After 20 seconds, the car's velocity is 10 mph to the left.
(Once again, we add to the speed because the acceleration and velocity are in the same direction.)
Let us summarize. The car's acceleration is 5 mph/4 seconds to the left. What this means is that its speed will change by 5 mph for every 4 seconds. Whether we add or subtract the 5 mph from the speed of the car depends on the relative direction between the velocity and the acceleration of the car, i.e., whether they are in the same or opposite directions.
During the first 12 seconds, the velocity and acceleration were in opposite directions. Therefore, we subtracted the change in speed caused by the acceleration from the speed of the car. In our case, the acceleration was 5 mph/4 seconds to the left. The change in speed was 5 mph, and we subtracted 5 mph from the speed of the object for every 4 seconds the car was undergoing the constant acceleration, as long as its velocity and acceleration were in opposite directions.
From 12 seconds to 20 seconds, the velocity and acceleration were in the same direction. Therefore, we added the change in speed caused by the acceleration to the speed of the car. Since the change in speed was 5 mph, we added 5 mph to the speed of the car for every 4 seconds the car was undergoing the constant acceleration, as long as its velocity and acceleration were in the same direction.
Next next time, I will introduce a formula that does all of this with much greater simplicity. I did not introduce it first because I felt it was more important to understand what was going on with the physics before learning to plug and chug. What the formula will serve to do is to illustrate the power and beauty of mathematics because it will allow us to solve all these problems without going through all this thinking.
However, before we talk about this wonderful formula, we'll need to take a detour (the first of many, as you shall see) and talk about something else. It won't go to waste because we will need to know this information before we can move on to the formula. In the mean time, take a break and think about some of the stuff we talked about here. Some of it is a little difficult and will probably need some time to sink in, especially given my poor communication skills. In particular, make sure you understand how to figure out the direction of the acceleration and what constant acceleration is, especially example 4 above. If you thoroughly understand these two topics, then you are doing well and should have no problems with the formula later on.
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