• Kinematics: In One Dimension and Under Constant Acceleration (continued)
  • Distance Formula
    • Choice of the origin
    • Solving for time
      • Example 1: Initial velocity and acceleration in the same direction
      • Example 2: Initial velocity and acceleration in opposite directions
• Next Page

5. Kinematics: In One Dimension And With Constant Acceleration (continued)

1. Distance Formula

   

   

   As the name of the formula implies, this formula is primarily used to find the distance an object has traveled if it is undergoing constant acceleration. Like the velocity formula, this is not its only purpose. It can be used to find any one of the quantities as long as all the other quantities have values for them. In other words, you can solve for initial distance, initial velocity, acceleration, or time if you have values for all of them except for the one you are solving for. For example, if you wanted to find the acceleration, you would have to know values for distance, initial distance, initial velocity, and time.

   Since you have had plenty of practice with the velocity formula, there is really no need to write down a procedure here for using the distance formula. The procedure is very similar, and it is probably ingrained in your mind already.

   Let me just give you a tip for using the distance formula. Since we are usually free to designate what the starting point or origin is, it is usually easiest to set it so that the initial distance is zero. In other words, we can use the initial position of the object under consideration as our starting point or origin. From now on, I will refer to the starting position as the origin. The origin is considered to be the zero. Distances measured to the right of the origin are considered positive because we are using the same sign convention as in the velocity formula above. Distances to the left of the origin are considered to be negative. In addition, please keep in mind that the distance and the initial distance are both measured from the origin and not from the initial position of the object under consideration.

   Most importantly, the answer you get from using the distance formula will be relative to your choice of origin.

   Before continuing on, let me summarize some important points about the origin.

   Origin
   
   1. The origin is the point from which you measure everything else. Specifically, both the initial distance and the distance are measured from the origin.

      Sign Convention: This sign convention is chosen to be consistent with the sign convention for the velocity formula.

      1. In the horizontal, distances to the right of the origin are (+), and distances to the left are (-)
      2. In the vertical, distances above the origin are (+), and distances below the origin are (-).

   2. The choice of the origin is arbitrary. You can choose the origin to be anywhere. In most of the problems you will encounter, the most helpful choice will be to make the origin coincide with the initial position of the object under consideration.
   3. By changing the origin, the physical situation should not change.

   This might be a little confusing, so let me give you an example to make the point clear.

   • Example 1
     How far does a car travel in 10 seconds if it undergoes a constant acceleration of 5 (m/sec)/sec to the right? Assume the car is initially at rest.

     We will solve this problem several ways. Just play along for now. There is a reason why I am doing this.

     • Method 1
       For the first time, let's be smart and call the initial position of the car the starting point or origin. Therefore, the initial distance is 0 meters because the initial position of the car is at our starting position or origin. Since it is always good to draw a picture, I have included one below.

       

       Putting in the following values, we will be able to get the correct answer. Notice that the sign for the acceleration is (+) because the acceleration is to the right.

       

       After plugging these values into the distance formula above, we find that the answer is x = +250 meters. Therefore, the answer is that the car is 250 meters away from its initial position after undergoing a constant acceleration of 5 (m/sec)/sec to the right for 10 seconds.

       However, you might ask, "Is it 250 meters to the right or left?." Well, in this problem, it is fairly easy because we know that it started out with an initial velocity of zero and that it accelerated to the right. Therefore, the car must have moved to the right a distance of 250 meters. However, there is an easier way of keeping track of the car. It involves looking at the sign of the answer that you get. When I wrote down the answer above, you might have wondered why I explicitly put down the (+) sign. The (+) sign was there for a reason. In fact, it serves to tell you whether the position is left or right of the origin. If the sign is (+), it is to the right of the origin. If the sign is (-), it is to the left of the origin. Therefore, since x = +250 meters, the car is 250 meters to the right of the origin after 10 seconds. Notice that x denotes the position as measured from the origin not the distance measured from the initial position of the car. In this example, they just coincide because we chose the initial position of the car as our origin.

       Let me ask one more question before trying to solve this problem in a different way. How far has the car moved from its initial position (not the origin) after 10 seconds? You will notice that this a slightly different question than asking how far has the car moved from the origin. However, since the initial position and the origin coincide in this problem, the answer to them both is the same. The answer is that the car has moved 250 meters to the right of its initial position after 10 seconds.

     • Method 2
       This might still be a little confusing, so let's solve this same problem but move the position of the origin away from the initial position of the car. In fact, for this part, let's move the origin to the left of the initial position of the car by 50 meters so they no longer coincide. Remember the origin is considered to be zero. Any distances to the right of it are (+), and any distances to the left of it are (-). Let me redraw the picture to better reflect this new choice of origin. You should note that these pictures are not drawn to scale.

       

       The important thing to note here is that the initial distance is also measured relative to the origin. Since the initial distance is 50 meters to the right of the origin, we have that the initial distance is +50 meters. Once again, the (+) sign is because the initial position of the car is to the right of the origin.

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       Next, try to solve for x without looking below. It will give you good practice. If you have any trouble, you can look below to see how to set up the problem properly.

       The answer should be that x = +300 meters.

       The correct values to put into the distance formula are as follows.

       

       You should have been able to easily get the values for the time, acceleration, and initial velocity. The only thing that was a little different might have been the initial distance. However, by following the sign convention and measuring the initial distance from the origin, you should have been able to get the correct value for the initial distance.

       So, let's try to interpret the answer we got. Physically, the answer should be the same as the first way of solving it because all we did was move the origin. This shouldn't affect the physical situation. Since the answer is x = +300 meters, the car is 300 meters to the right of the origin after 10 seconds. On the surface, this seems to be a different answer than what we got from the first way of solving it because the answer we got above was that the car was 250 meters to the right of the origin. However, since the origin is something that we can assign to be anywhere, the distance as measured from the origin is not a good thing to compare because we can always change the answer by moving the origin. The question that we should be asking is how far the car has moved from its initial position. This shouldn't change because there is only one correct answer for this. Remember the math that we do should reflect what happens in the real world. By simply changing the origin, we shouldn't be able to change the physical situation. Since the car's initial distance is 50 meters to the right of the origin and the car's final distance is 300 meters to the right of the origin, the car's final distance is 250 meters to the right of its initial position. If you will compare this to what we got above, you will notice the answers are identical. In both cases, after 10 seconds, the car has moved a distance of 250 meters to the right of its initial position.

       It should be apparent now why choosing the initial position of the car to coincide with the origin is a good thing to do in this problem. It is because it saves us from having to go through some extra mathematics at the end to figure out how far the car has moved from its initial position, which was the answer we were after anyways. By choosing the origin to coincide with the initial position of the car, the answer we get will give us the distance the car has moved away from its initial position directly without further work. In addition, by looking at the sign and following the sign convention, it will also tell us whether the final position is to the right or left of the initial position.

       So, let me summarize a couple of notes here.

       • The choice of the origin is arbitrary. You can put it anywhere that is convenient. In most cases, the most convenient location to place the origin is to put it at the starting (or initial) position of the object under consideration so that the origin and the initial position of the object coincide.
       • The choice of the origin should not affect the physical situation.

     • Method 3
       For the final way of solving this problem, let's just make a really bad choice of location for the origin. This example should really drive home the usefulness of choosing a good origin. For this one, let's choose the origin so the initial position of the car is to the left of the origin by 50 meters.

       Since you've already done two of these already, try doing this one by yourself. Remember that the answer you get for the final distance, x, will be different from the answers you got above because the choice of the origin is different in all three. However, the physical situation should not change. Therefore, you should still find that the car is 250 meters to the right of its initial position after 10 seconds. In addition, be very careful with the sign when you put the sign for the initial distance. Remember that in the sign convention we are using, distances to the right of the origin are (+), and distances to the left of the origin are (-). Since pictures are always helpful, you should draw your own picture and then use it as a guide for what sign to give to the initial distance.

       After doing all of this, the answer is that x = +200 meters. If you are having a lot of trouble getting this answer, please refer to this section for a picture and a hint on how to set up the problem. As predicted, this answer differs from the answers we got above. However, you should still find that the car's final position is 250 meters to the right of its initial position. As a test for yourself, see if you can find the reason why this is still true before reading the next paragraph. Go back to the picture you just drew if you are having troubles with this.

       It is not too difficult to see why the car is still 250 meters to the right of its initial position. Since x = +200 meters, the car's final position is 200 meters to the right of the origin. However, since the car's initial position is 50 meters to the left of the origin, the distance between the car's final position and its initial position is 250 meters, with the car's final position being to the right of its initial position.

     Well, I hope this has given you enough practice so you are comfortable with using the distance formula. The only other way of making this problem more difficult is to introduce a non-zero initial velocity. However, this really shouldn't pose much of a problem as long as you obey the sign convention we agreeded upon. The next example involves a situation in which the initial velocity is not zero.

   • Example 2
     Assume a ball has been thrown upward with an initial velocity of 20 m/sec. While it is in the air, it experiences a constant downward acceleration of 10 (m/sec)/sec. What is the position of the ball after 2 seconds? The answer should be specific, including the distance and whether the ball is above or below its initial position.

     You already have everything you need to solve this problem. Make sure to draw a picture and use the correct sign convention. Remember that you can choose the origin to be anywhere, but a useful place would be to make the origin coincide with the initial position of the ball.

     The answer you should get should be that the ball is 20 meters above its initial position after 2 seconds. Remember that you should get this answer regardless of your choice of origins. If you did not get this answer, make sure you have the correct signs for the initial velocity and the acceleration.

     As a review, find the velocity of the ball after 2 seconds. You should find that the velocity is zero.

   Solving for time

   Like the velocity formula on the previous page, the purpose of the distance formula isn't just for solving for the distance. As you may recall, we should be able to theoretically solve for any one of the quantities in a formula as long as we know values for all the other quantities except for the one we are solving for. If you will recall, there are five different quantities in the distance formula.

   

   

   We have already covered how to solve for the distance. The initial distance is not really something we need to solve for because we are usually given this. In any event, we can always choose the origin to coincide with the initial starting position of the object under consideration thereby making the initial distance zero. In addition, the initial velocity and the acceleration are almost always given as well, therefore there is usually no need to solve for them.

   This leaves us with the time. A typical problem might ask how long it takes for an object to travel a certain distance. For example, a problem might ask how long it takes a car to travel 5 meters if it is initially at rest and its acceleration is a constant 10 (m/sec)/sec to the right.

   In its current form, the distance formula is not very helpful in solving for time, but with a little algebraic manipulation, we can put it into a more useful form. I will not put the derivation here but, let me just say that it involves the quadratic formula. If you are not familiar with the quadratic formula, don't worry about it. The result listed below is all you need for this section.

   

   

   In this current form, the distance formula is more suited to solve for the time. There is one thing to point out here. If you look at the formula, you will see that there is a + sign. This means there are two possible solutions when you use this formula. If you explicitly write it out, the two possible solutions for time are as follows.

   

   and

   

   The + sign tells us that one solution uses the (+) sign and the other solution uses the (-) sign. It might look as if it is just one formula, but the + sign tells us that it is two formulae. To summarize, in general, when you solve for time using the distance formula, you will get two solutions for the time.

   To get a better idea of how to use this formula to solve for time, I will give two examples. The two examples will serve to illustrate two common cases you might encounter when applying this formula. The first example will involve a case where the initial velocity and the acceleration are in the same direction. Recall that the acceleration and the net force are always in the same direction, therefore the first example will involve a case where the initial velocity and the net force are in the same direction. The second example will involve the case where the initial velocity and the acceleration (and the net force) are in opposite directions.

   • Example 1: When the initial velocity and the acceleration are in the same direction.
     Consider throwing a ball downward with an initial velocity of 5 m/sec. In addition, there is a downward constant acceleration of 10(m/sec)/sec. How long does it take for the ball to reach a position 10 meters below its initial position?

     Once again, before we start the problem, let us consider the physical situation. We have already encountered this situation many times before. Whenever the initial velocity and the acceleration (and net force) are in the same direction, the object will continue to speed up in the same direction it was initially moving in. Refer to this section if you need a reminder.

     Therefore, after we throw the ball downward, it will continue to speed up while moving downward because the net force acting on the ball is producing a constant downward acceleration on the ball. Assuming there is nothing below the ball to stop its motion, it will continue to travel downward, all the while speeding up. Therefore, there should only be one correct answer for this problem. At a certain time, which we will solve for, the ball will be at a position 10 meters below its initial position. After that time passes, the ball will have moved further than 10 meters from its initial position.

     Physically, we see there can only be one solution to this problem. However, you might have cleverly noticed that the formula we are using gives us two solutions. Could there be something wrong with the formula? Well, let's bravely push on and solve this problem and see what happens. We know there should only be one valid answer even though the formula gives us two solutions. Since we know this will occur, let's just keep our guards up while we solve this problem.

     Solving this problem is not really that exciting. We know we can use this formula because the condition of a constant acceleration is met. All that remains is to put in the values for the other variables besides time.

     • The initial velocity is -5 m/sec. The (-) sign is because the ball is initially thrown downward.
     • The acceleration is -10(m/sec)/sec. Once again, it is (-) because the acceleration is downward.
     • As for the initial position and the final position after the ball accelerates, these values will depend on your choice of origin. In this case, let's choose the origin to coincide with the initial position of the ball. In other words, we are choosing the initial position to be zero. If the choice of origin coincides with the initial position of the ball, what will be the final position of the ball after it accelerates over a distance 10 meters? Try to figure this out before looking below. It will give you good practice. Remember that the ball's position after it accelerates is 10 meters below its initial position.

       The correct value for x should be x = -10 meters. The (-) sign is because the ball is 10 meters below the origin. If the ball were 10 meters above the origin, then it would be (+). If you need to review the sign conventions, go to this section.

     Since we know all the other values except for t, we can use the formula to solve for t. There will be two solutions. One will correspond to the (+) solution in the formula, and the other will correspond to the (-) solution in the formula.
     • The solution using the (+) is t = -2 seconds.
     • The solution using the (-) is t = 1 second.

     {As an aside, my thanks to James J. O'Connor for pointing out a mistake in the answer above when this problem was originally posted. It is now correct thanks to his attention.}

     As expected, we get two solutions: one from using the (+) in the formula and the other from using the (-) in the formula.

     However, we should be more careful with the interpretations of the solutions. While the second solution of t = 1 second seems reasonable, the first solution of t = -2 seconds does not seem reasonable. It does not seem reasonable because we know time moves in only one direction, therefore the (+) time solution of t = -2 seconds is not a valid solution. Because time moves forward, we know that only solutions for the time, which are greater than zero, are valid, while those less than zero are not valid. This is because solutions for the time, which are less than zero, would correspond to moving backwards in time.

     So, you see, we were justified in keeping our guards up. Even before solving this problem, we knew there would be two possible solutions for the time, however, we also knew that, physically, there should only be one valid solution. Instead of blindly plugging along and accepting both solutions as valid, we knew only one solution would be valid. This is an important idea to keep with you as you study physics. It is not enough to get an answer, but you really need to think if the solution is a valid answer. In addition, it is also important to get a feeling of what is physically going on in the problem before solving it. In this example, we knew the ball was falling down. Because of this, we knew there would only be one time at which the ball would be 10 meters below its initial position. After that moment, the ball would continue to fall and never come back up. Because we knew this beforehand, we were able to predict that only one of the time solutions would be valid.

     Therefore, when you encounter a case where the initial velocity and the acceleration are in the same direction, there should only be one correct answer which corresponds to the correct flow of time.

   • Example 2: When the initial velocity and the acceleration are in opposite directions
     Consider throwing the ball initially upward at 20 m/sec. In addition, consider the case where there is a constant downward acceleration of 10 (m/sec)/sec. How long will it take for the ball to reach a position 15 meters above its initial position?

     Before we solve this problem, let us consider the situation first. We are throwing the ball upward initially with a constant downward acceleration. Therefore, the initial velocity and the acceleration are in opposite directions. Since the ball is experiencing an acceleration, there must be an external net force which is responsible for this. Because the net force causes the acceleration, the net force and the acceleration are in the same direction. As a result, the net force and the initial velocity of the ball are also in opposite directions. You might be inclined to inquire as to the source of this force. That is a very good question but let's ignore it for now. We will get to this later. We have already encountered this situation before in which the net force and the initial velocity are in opposite directions. You should already know qualitatively what happens to the ball. The ball should slow down as it moves upward. It will eventually slow down to 0 m/sec. After it reaches this point, the ball will then start to speed up while moving downward. If you need to refresh your memory, please refer to this section.

     Next, let's just plug away and see what we get. The following are the correct values you will need to plug into the formula above in order to get the correct answers. Try to get the correct values on your own and then compare them with the values below.

     • The initial velocity is +20 m/sec. The (+) sign is because we threw the ball up.
     • The acceleration is -10 (m/sec)/sec because the constant acceleration is downward.
     • Once again, we are free to choose the origin to be convenient for the problem at hand. In this case, let's just choose the initial position of the ball as our origin. Since the position of the ball is 15 meters above the initial position of the ball after it accelerates, the correct value we should use is x = +15 meters. The (+) sign is because the position of the ball (after it accelerates) is above the origin which we have chosen to coincide with the initial position of the ball.

     Once again, we will obtain two solutions. One solution occurs when you use the (+) sign. The other solution occurs when you use the (-) sign.

     • The solution using the (+) sign is t = 1 second.
     • The solution using the (-) sign is t = 3 seconds.

     Uh oh. What happened? In the previous example, we could rule out one of the solutions by reasoning that time does not go backwards. However, we cannot use that line of thinking to rule out one of these answers because both times in this example are positive, so they both correspond to the normal flow of time. If we cannot rule out one of these answers, can both of them be correct? Hmm..... Now would be a good time to reconsider what is happening to the ball as it moves. Think about how the ball travels after you throw it up. Drawing a picture here might help. Here's one hint: In addition to thinking about what happens to the ball as it rises, consider what the ball does as it falls back down. Take some time to ponder this before reading further.

     Well, let's draw a picture of what's going on in this example. Maybe that will help us decide whether only one of the two answers is correct or if both are correct. Instead of drawing a generic picture, let's be a little more specific. In order to do this, let us calculate the velocity and position of the ball as it travels through the air. To be specific, let us consider the velocity and position of the ball for each second the ball is in the air. Using the velocity formula and distance formula above, we can calculate the velocity and position of the ball for each second the ball is in the air. The data is presented in the table below. You should have no problem calculating the velocity and position of the ball because you have already done this above.

     Velocity and Position of the Ball in Flight

     Time	Velocity	Position	Comments
     t = 0 sec	v = +20 m/sec	x = 0 meters	This is the initial position of the ball at t = 0 seconds. We have chosen the initial position as the origin. No calculation needed here because all this data was given as part of the problem.
     t = 1 sec	v = +10 m/sec	x = +15 meters	After 1 second, the ball has slowed down while moving up 15 meters from its initial position. The ball is still moving upward because its velocity is (+). It is above the initial position because the distance is (+).
     t = 2 sec	v = 0 m/sec	x = +20 meters	After 2 seconds, the ball has slowed down to 0 m/sec. The ball is momentarily at rest only at this exact time. This corresponds to the peak because it is the highest point the ball reaches. After this point, the ball will begin to move downward and speed up.
     t = 3 sec	v = -10 m/sec	x = +15 meters	After 3 seconds, the ball is moving downward while speeding up. It will continue to move downward while speeding up. The ball is moving downward because the velocity is (-). The ball is still above the origin (which we have chosen to be its initial position in this case) because the position is still (+).
     t = 4 sec	v = -20 m/sec	x = 0 meters	After 4 seconds, the ball has returned to its initial position. However, it is now traveling downward instead of upward. Notice that the speed is the same as when the ball was initially thrown upward. However, the velocity is different because the ball is now traveling downward, while it was traveling upward initially.
     t = 5 sec	v = -30 m/sec	x = -25 meters	After 5 seconds, the ball is moving downward and is below its initial position. It is below its initial position because the position is (-).

     Now, that we have the data we need, let's draw the picture. You might have noticed that I stopped after 5 seconds. The ball does not stop after 5 seconds. In fact, it will continue to move downward and pick up speed. I only stopped it here because of convenience.

     

     This is a picture of the ball while it is in flight. We only did it for 5 seconds of flight because of lack of space. The picture is a little inaccurate. In the picture, the ball seems to curve around top as it reaches the peak. In reality, the motion of the ball should be straight up and down, with no curving at all near the peak. The picture was drawn this way for clarity. Remember, all these examples are in one dimension, so the movement can only be in a straight line.

     If you look carefully at the picture, you will notice that the ball slows down as it rises up toward the peak. This is as expected because, during this time, the velocity of the ball is opposite to the direction of the net force which is producing the constant downward acceleration on the ball. After the ball reaches the peak, you should notice that the ball begins to speed up while moving downward. This is also no surprise because, during this time, the velocity of the ball is in the same direction as the net force which is causing a constant downward acceleration on the ball.

     There are a couple of things to notice here in this example. These remarks will hold true in general as well.

     • First, there is a symmetry to the motion of the ball. As the ball moves up and down, there are times when the ball has the same speed. For instance, at t = 1 sec and t = 3 sec, the ball has the same speed of 10 m/sec. The difference is that at t = 1 sec, the ball is moving upward at 10 m/sec while at t = 3 sec, the ball is moving downward at 10 m/sec. Notice, also, that at the times when the ball is going at the same speed, the ball is also at the same height. You will also notice a similar thing happening to the ball at t = 0 sec and t = 4 sec. Finally, keep in mind that, while the speeds may be the same, the velocities are not the same because of the difference in direction.
     • Second, the ball is the highest when it is at the peak, which coincides with the moment the ball is at rest. In this case, this occurs 2 seconds after the ball is thrown up.

     So, what was the point in all of this? Oh yeah, I remember. We were trying to figure out whether both answers to example 2 above were correct or if only one answer was correct. Well, indeed, we have solved that. At t = 1 sec, the ball is at 15 meters above its initial position. Likewise, at t = 3 sec, the ball is at 15 meters above its initial position. Both answers are indeed correct. The difference is that at t = 1 sec, the ball is moving upward while at t = 3 sec, the ball is moving downward. The only difference is the direction of the velocity. Both the speed and position are identical. Since the problem did not differentiate between whether the ball was moving up or down, both answers are valid and correct. If the question had asked at what time was the ball moving downward and at a position 15 meters above its initial position, only the answer of t = 3 sec would be correct. However, that was not the question asked.

     Therefore, when you come across a case where the acceleration and initial velocity are in opposite directions, be prepared for the possibility that two solutions may be correct. Now that we are aware of a possibility in both answers being correct, does this mean that you should always expect two correct solutions whenever you see a situation similar to this? Be careful here before you answer. See if you can come up with some counterexamples before reading further.

     The answer is that you do not always get two valid answers. One counterexample already presented itself to us above. If the question was more specific, we could rule out one of the two solutions. For instance, if the question asked for the time when the ball was moving downward at a position of 15 meters above its initial position, there is only one valid answer. If this was the question, the answer of t = 1 sec would not be a valid answer. However, there is another counterexample we can come up with. Hint: Look carefully at the picture above and consider what is happening to the ball.

     Specifically, look at the time, t = 5 sec. At this time, the ball is moving downward at 30 m/sec, and its position is 25 meters below the initial position of the ball. Looking carefully at the picture, is there another time at which the ball is at 25 meters below its initial position? The answer is no. After t = 5 sec, the ball continues to move downward. It never comes back up after this point, so there is no hope of the ball ever coming back up to the same position again after t = 5 sec. Likewise, if you look at the times before t = 5 sec, there is no time at which the ball is ever 25 meters below its initial position. Therefore, in this case, there can only be one valid answer. The other one must be incorrect.

     You can prove this last point to yourself by solving example 2 above again but with a slight change. Instead of 15 meters above its initial position, solve for the time when the ball is 25 meters below its initial position. As you already know from the picture, one answer should be t = 5 sec. This is the correct answer. Therefore, the other solution you obtain should be an invalid solution because we have already determined that there can only be one valid answer. Using this formula, you should obtain the following solutions.

     • Using the (+) formula, the solution is t = -1 sec.
     • Using the (-) formula, the solution is t = +5 sec.

     As we have already discussed, the t = +5 sec is the valid solution because it corresponds with the correct forward flow of time. We can also see this is the correct answer from the picture above. The t = -1 sec is not a valid solution because it is less than zero and corresponds with a backward flow of time.

   So, why did we really spend all this time with these examples? Well, it actually serves to bring a lot of what we have been talking about together. First, it gives us a clearer picture of what is going on when there is a net force acting on an object. Specifically, we were able to calculate the position and velocity of an object under the influence of a net force which was causing the object to undergo a constant acceleration. Previously, we were only able to determine whether the object would speed up or slow down, but here we were able to give actual numbers. We were actually able to determine both the velocity and position of the object. Second, it drives home the message of why we bother with the acceleration at all. Without knowledge of the acceleration, we would not be able to predict the motion of the ball while it was in the air. Third, we saw how drawing pictures can help make these problems a little easier to understand and visualize. Finally, we saw the importance of really looking at the answer and asking if it was the answer we were expecting or if there is something amiss. Once again, it is not sufficient to get any old answer. It is imperative that the answer we obtain matches up with what happens in the real world.

   Okay, well that was a lot of clumsy information to lump on one page. If you went through this in one sitting, you must be extremely tired and bored by now. In any event, you have earned a well-deserved break. Take some time off and if something wasn't clear, try thinking about it some more and going over some of the examples. Next time, we will learn to use some of these formulae in combination with one another.