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

  1. Using the Distance and Velocity Formulae in Combination: Bet You Can't Just Use One

    Up until this point, we have just been using one formula for each problem we have come across. In an ideal world (some would say "lazy"), that would be all we would need, however, in the real world, things just don't happen so easily. Sometimes, you will just have to use more than one formula to solve a problem you come across. And, like in those Streetfighter games, they become more powerful when you learn to use them in a combo. (As an aside, I've never been able to consistently pull off those combos in Streetfighter.) So, without further ado, let me give you an example to illustrate the point.

    • Example: If you throw a ball upward with an initial speed of 20 m/sec, what is the height of the ball at its peak? Assume the ball experiences a constant downward acceleration of 10 (m/sec)/sec.

      Once again, you might be wondering what is this force which is causing the constant downward acceleration of 10 (m/sec)/sec. You have every right to ask, and I promise we will get to it soon enough. Wow, I'm feeling more and more like Papa Smurf everytime I bring this up.

      Now, you might have observed that this problem is almost like the last problem we came across. If you saw this, you might have just flipped back to take a look at the picture. Problem solved. Wow, that was easy. Well, you could have very well done this, but that's not the point. We are trying to learn to use the formulae we know in combination with one another.

      Before going further, it would be a very good idea for you to try to figure this problem out on your own. You already have all the knowledge you need. All that is required is some detective work. Try to do it without looking at the following hint first. Look at it only after you really get stuck. (Hint: You need to use the velocity formula and the distance formula exactly once each. Be careful, however, you might have to use them in a different form.)

      Finally, don't forget to draw a picture because that usually helps.

        Solution
      1. Okay, let's get started with the problem. First, let's write down what is given to us by the problem. Go ahead and do this on your own and then compare your results with the data below. Be sure to write everything down.

        Name of Known Quantity Value Given Comments
        Constant Acceleration -10 (m/sec)/sec The sign is (-) because the acceleration is downward.
        Initial Velocity +20 m/sec The sign is (+) because the ball was initially thrown upward.

        Hmmm..... Are these all the quantities we were given? It certainly seems like it, but there must be something missing otherwise I would have never asked that question. Well, we actually know one more quantity. If you've already figured this out, give yourself a pat on the back. If you haven't, think about what happens to the ball in the problem. Let's see. We throw the ball upward at 20 m/sec, and we want to know its height when it is at its peak. Well, we can rule out that we were given the height of the ball at its peak because if that was given, then the problem would be trivial. Also, we were not given the time it takes for the ball to reach its peak. Well, how about the final velocity? Hmmmmm.....

        Well, on the surface, it doesn't look like we were given that either. But, wait. When the ball reaches its peak, that's the highest it will go. After it reaches the peak, the ball starts to speed up while falling back down. On the way up to the peak, the ball slows down because its velocity is in a direction opposite to the net force. Therefore, at the peak, the ball is momentarily at rest. So, it turns out that we do know the final velocity of the ball.

        Name of Known Quantity Value Given Comments
        Final Velocity 0 m/sec At the peak, the ball is momentarily at rest, so its speed is 0 m/sec.

      2. Now we have written down all the known quantities, let's try to solve this problem. Well, since we want to solve for the height of the ball at its peak, we should probably use the distance formula.

        This formula certainly seems to fit the bill. Our current problem satisfies the condition for using this formula because the ball is undergoing a constant acceleration. And, this formula can be used to solve for distance as we have done before.

        For the choice of origin, we can just choose it to be the initial position of the ball. Now that we have made that choice, we can proceed to solve the problem. We know the acceleration and the initial velocity, but we don't know the time. Man, that was so close. The distance formula certainly seemed like the correct formula to use, but it looks like we weren't given all the information we needed to solve this problem because we are missing the time.

        If only there was some way to figure out how long it takes for the ball to reach its peak, then we would be home free. It certainly seems hopeless because the distance formula appeared to be the only formula we could use that would allow us to solve for distance.

      3. Since we have gotten as far as we can with the distance formula, let's take a look at the velocity formula and see if we can get lucky. Now, we obviously can't use the velocity formula to directly solve for the height of the ball at its peak, but let's see if we can use it to solve for something that we need to finish this problem.

        First, let's check to see if we can even use this formula in our problem. Once again, since our problem involves an object undergoing constant acceleration, the condition for using this formula is satisfied. Next, let's see if there is any way we can use this formula to solve for the time it takes the ball to reach its peak. Because if we can solve for the time it takes the ball to reach its peak, we can use this data and put it back into the distance formula along with the other known quantities (initial velocity and acceleration) to solve for the height of the ball at its peak. Remember, because we were missing the time it takes the ball to reach its peak, we couldn't use the distance formula to solve for the height of the ball at its peak. The time it takes the ball to reach its peak was the only missing piece of information we needed in order to use the distance formula to solve for the height of the ball at its peak.

        Next, recall that we can rewrite the velocity formula in a form which is easier for solving for the time.

        If you need a reminder, go back to this section. Well, it looks like this might work if we know the final velocity, initial velocity, and acceleration. But, these are precisely the quantities we were given in the problem.

        By putting in these values, we see that it takes t = 2 sec for the ball to reach its peak, which corresponds with the time it takes for the ball to slow down to 0 m/sec while moving upward.

      4. Finally, since we have figured out that it takes the ball 2 seconds to reach its peak, we can plug this value into the distance formula above along with the other given quantities to figure out how high the ball is at its peak.

        After putting in the values, the answer you should arrive at is +20 meters. Remember that it is very important to note the sign because it tells you whether the position is above or below the origin. As expected, the answer is (+) because, at its peak, the ball should be above the initial position of the ball, which we chose to be its origin.

        Here's the picture again for clarity. Once again, the ball's motion should be straight up and down, however, the picture was drawn this way so things wouldn't get crammed up. The ball's motion does not arc near its peak.

      Let's summarize what we just did and see if we can learn something useful from the example above. The problem asked us to solve for the height of the ball at its peak. First, we looked at the problem and wrote down what was given. This is a very important first step because we don't have to go solving for this data later on. Why do extra work when someone has already provided us the information? Also, we saw that writing down the given information is not always so easy. For example, it was not explicitly given that the final velocity of the ball was 0 m/sec, but it was implied because we were looking for the height of the ball at its peak. As a result, because we know the ball's velocity at its peak is 0 m/sec, we were able to determine the final velocity to be 0 m/sec. So, the hint here is don't rush into the problem. First, take some time to think about what was given as part of the problem and see what kind of useful information is given.

      Next, we looked for a formula that satisfied the conditions of the problem and solved for the answer we were after. In the case above, we were after the height of the ball at its peak. As a result, we correctly chose the distance formula because it solves for the height and satisfies the condition that the object under consideration must be undergoing a constant acceleration.

      However, at this point, we ran into a problem. The formula we had chosen didn't seem to work because we were missing some information. Specifically, we were missing the time it takes the ball to reach its peak. This was the key missing piece of information we needed in order to make the distance formula work because we knew everything else (except for the distance, of course). At this point, it is important to make a note of what information we know and what we are missing. Because we knew the initial velocity and acceleration, we did not have to solve for them. In addition, since we knew we were missing the time it takes the ball to reach the peak, we were able to isolate what we needed in order to solve the problem.

      By figuring out that we needed the time, we were able to recast the problem in a different light. It showed us a new direction to pursue because if we found the time, we could then use it to find the height. It was at this point that we decided to look at the velocity formula to see if it would yield anything useful. This turned out to be the turning point in our battle with this problem, so let's take a closer look at why we chose to use this formula. In hindsight, we can see that we used this formula because it allowed us to solve for the time it takes the ball to reach its peak. However, there was another reason why we turned to the velocity formula.

      In fact, we had to turn to the velocity formula because the distance formula took us as far as it could take us. What do I mean by this? Well, if you will recall, the distance formula can also be rewritten in a different form which allows us to solve for the time. So, why did we use the velocity formula instead of the distance formula to solve for the time? Think about this for a moment before reading further.

      To figure out the reason why we couldn't use the distance formula to solve for the time, think about what you would need to know in order to solve for the time using the distance formula. In other words, what quantities would we have to know in order to use the distance formula to solve for time? You would need to know the initial velocity, acceleration, and the distance. While the first two were given to us, the distance was not. In fact, the distance is precisely the quantity we are solving for. And, if we had known this, there would have been no reason to do this problem. So, if we had tried to use the distance formula to solve for the time, we would have been stuck in a circle: to solve for the distance, you would need the time and to solve for the time, you would need to use the distance. This is a never-ending circle (refer to the picture below). The only way out is to use a different formula. That is why we turned to the velocity formula. Therefore, in general, if you have more than one unknown variable in a formula, you must use a different formula to figure out all but one of the missing variables. You cannot use the same formula, even in its different forms, to solve for more than one unknown variable because if you do, you will end up in a circular argument.

      Circular Argument: Using the same formula (even in its different forms) when missing more than one variable.
      Use the distance formula, , to solve for the height of the ball at its peak. Assume initial velocity and acceleration are given. To do this, we need to know the time it takes the ball to reach its peak. Let's try to figure out the time using the distance formula in a different form. Use the distance formula in a different form, , to solve for the time it takes the ball to reach its peak. To do this, we need to know the height of the ball at its peak. However, that's precisely what we want to solve for. As a result, we are stuck in a circular argument that gets us nowhere because, to solve for the height, we have to go back to the left and use the distance formula in its original form.

      Finally, after using the velocity formula to figure out the time the ball takes to reach its peak, we were able to use the distance formula to find the height of the ball at its peak. Let me make a couple of more points here. If you will recall, all the formulae we used satisfied the conditions of the problem. If they did not, then our solution would not be valid. For example, the problem above involved a ball undergoing a constant acceleration. Both the velocity and distance formulae we used above satisfied this condition, so our solution is valid. In addition, don't give up when solving a problem. If the problem was given, there is a solution. You might have to try something new and think a little differently, but there is a solution.

        Main Points To Remember:

      1. You cannot use the same formula (even if it is in a different form) to solve for more than one variable. If you find yourself in a situation where the formula you want to use has more than one unknown variable, you will have to turn to a different formula or formulae to solve for all but one of the unknown variables.
        In our example above, we wanted to use the distance formula to solve for the height of the ball at the peak. However, we could not solve for the height because we were missing the time it takes the ball to reach the peak. As a result, we had to turn to a different formula to solve for the missing time, so we could then use that information to go back and solve for the height. In this case, we turned to a different form of the velocity formula to solve for the time. If we had tried to solve for the time using a different form of the distance formula, we would have gone nowhere because we cannot use the same formula (even in its different forms) to solve for more than one unknown variable. This is an extremely important point to remember, especially if this is your first time learning physics. You'll be surprised to see how many students crack under the pressure of an exam and forget this little tidbit during an exam.

      2. Every formula you use must satisfy the conditions of the problem. In our example above, all the formulae we used to solve the problem satisfied the condition that they were consistent with an object undergoing constant acceleration. Even if we were able to get a solution using a formula inconsistent with the condition of constant acceleration, the solution would not be a valid one. Once again, this shows the importance of getting a valid solution, consistent with the physical world, rather than just getting any old solution. This is another very important point that some students forget under the stress of an exam.

      3. Finally, don't give up. Even if you have to take a break from the problem and come back to it later, don't give up. Problems are usually posed for you to learn something new or to test if you have learned what the instructor has just taught you, so there is usually a solution. View the problems as a challenge and really try to tackle them before asking for help. On the other hand, don't ever feel too embarrassed to ask for help. There is no shame in asking for help especially when learning something new and strange.

      Well, I hope this last discussion was clear. I might have been overly repetitive, but it was only because these points are very important.

    Now that we have just seen one example of what can be done when we use more than one formula to solve a problem, I'm sure you can imagine a couple of other examples involving different ways to use the velocity and distance formulae (in their different forms) to solve problems. In fact, it is a very good idea for you to make up your own problems to see how these two formulae can be used in combination with one another. It will give you an idea of what these formulae can and cannot do. And, if you take a physics course in the future, it will give you an idea of some of the problems to expect.

    Take some time now to come up with some problems on your own that are of a different variety than the one above. If you run into difficulty, the following are some example problems along with solutions for them. However, try your best to make some up on your own first.

    Some Useful Examples:
    For all of the following examples, I will assume there is a net force on the ball which is causing the ball to experience a constant downward acceleration of 10 (m/sec)/sec. Surprise! Surprise!

    Try working through the examples by yourself without looking at the hints. Also, don't forget to think about what the ball is physically doing before solving the problems.

    • Example 1:
      Release a ball initially at rest. What is the velocity of the ball when it reaches a point 20 meters below its initial position? (Assume the ball experiences a constant downward acceleration of 10 (m/sec)/sec.)

      The answer you should get is that the ball velocity's is 20 m/sec downward when it is 20 meters below its initial position after being released from rest. Try to get this answer on your own before looking at the hint below.

      Hint: In order to solve this problem, you will need to use the following formulae.
      and

      To solve this problem, we need to use the velocity formula. However, while we know the initial velocity and the acceleration, we do not know how long it takes the ball to reach a position 20 meters below its initial position. That is why we used a different form of the distance formula to solve for the time first. In addition, only one of the two time solutions will be a valid solution. Refer to this section if you need a reminder about this. If you had trouble with this problem, these hints should steer you to the correct solution.

    • Example 2:
      Consider the following situation. You are sitting on top of a tree 15 meters off the ground. For some unknown reason, you throw a ball upward. After 4 seconds, the ball is moving downward at a speed of 20 m/sec. What is the position of the ball relative to the ground? Once again, assume the ball experiences a constant downward acceleration of 10 (m/sec)/sec.

      The answer you should get is that the ball is 15 meters above the ground after 4 seconds. This problem is a little tricky, so if you get the wrong answer, try drawing a picture and be careful when choosing the origin.

      Hint: In order to solve this problem, you will need to use the following formulae.
      and

      To solve this problem, we need to use the distance formula. However, while we know the time and the acceleration, we are missing the initial velocity. As a result, we need to solve for the initial velocity first. This is why we turned to a different form of the velocity formula to solve for the initial velocity first before using the distance formula. Once we solved for the initial velocity, we can then put this into the distance formula along with the time and the acceleration in order to solve the problem.

      Now, there was a little trick in this problem. Normally, we choose the initial position of the object to correspond with the origin. However, in this case, a better choice is to pick the origin to be the ground. Was there a reason why we did this? Well, it is because the answer we wanted is the height of the ball relative to the ground. If you chose the ground as your origin, the answer you get from the distance formula is the answer to the problem because the ground and the origin are one and the same. We would not have to shift the solution to the distance formula in order to get the answer. Remember, the solution to the distance formula is always with relation to the choice of origin.

      If you had chosen the initial position of the ball to be the origin, the solution you get from the distance formula would be 0 meters because, after 4 seconds, the ball is back at its initial position. However, this would not have been the correct solution to the problem because the problem asked for the position relative to the ground, not relative to the initial position of the ball. To get the correct solution, you would have to recall that the initial position of the ball was 15 meters above the ground, therefore, after 4 seconds, the ball has returned to its initial position which is 15 meters above the ground. Once again, we see the importance of choosing an origin and also the importance of keeping track of where we have defined the origin to be.

      This problem could have been solved by either choice of origin, but it was slightly more convenient to choose the origin to be the ground. If you are still unsure about how to choose the origin, please refer to this section.

  2. Making contact with Newton's First Law of Motion

    Well, it's been awhile since we've talked about Newton's First Law of Motion, so let's see what it has to do with this section.

    First, let's recall Newton's First Law of Motion. Simply stated, an object's velocity (both speed and direction) will not change unless the object is acted upon by a nonzero external net force. In other words, an object will continue to move at the same speed and in a straight line unless a nonzero net force acts upon it. This is because, if there is no net force acting on an object, the object does not experience an acceleration. As a result, its velocity does not change because, by definition, an acceleration is a change in velocity. Therefore, when an object does not experience an acceleration, its velocity does not change.

    Let's take a look at the velocity formula and see what happens in a case where the object does not feel a net force. Here is the velocity formula again.

    Given the condition that there is no net force acting on the object, what is implied about the object's acceleration? Well, since force causes an object to accelerate, a zero net force means the object is experiencing no acceleration. Therefore, a = 0 (m/sec)/sec. Plugging this into the velocity formula, we arrive at the following expected result.

    It is important to remember that this formula only holds when the net force is zero.

    This formula tells us that, when the net force is zero, the object's final velocity is the same as its initial velocity. In other words, the object's velocity does not change when there is no net force. This is precisely what Newton's First Law of Motion tells us, so our velocity formula is consistent with Newton's First Law of Motion.

    So, if an object is initially at rest, it will remain at rest unless disturbed by a net force. Likewise, if an object is moving in a straight line at 55mph, it will not deviate from that line or speed unless disturbed by a net force.

    Okay, well that's all and good, but how does this translate into the motion of the object? Basically, we are asking what does the motion of an object with a constant velocity look like? We sort of already know this. Because we know constant velocity means constant speed and direction, the object's speed does not change and it must continue to move in a straight line and not deviate from that line. But, what about how much distance the object travels over time? You probably already know this, but I'll give a couple of examples nonetheless.

    Consider an object moving at a constant speed of 10 meters/sec in a straight line. What this means is, for every second, the object travels 10 meters. Likewise, an object moving in a straight line at 55 mph travels 55 miles for every hour. What about an object moving at a speed of 15 meters/2 seconds in a straight line? This just means it takes the object 2 seconds to travel a distance of 15 meters. I told you it wasn't hard.

    So, to sum it all up, given the condition that there is no net force acting on the object, our velocity formula predicts that the object's velocity will not change over time. The object will continue to move in a straight line and at the same speed.

    Isn't it comforting that our velocity formula is consistent with Newton's First Law of Motion? I know, I for one, will sleep better tonight knowing this.

    So, what was the point of all this? Well, it was just to show you that the mathematical formulae we use are consistent with the real world. For instance, in this section, we showed that the velocity formula is consistent with Newton's First Law of Motion. If the velocity formula was not consistent with the real world, then we should be wary of using it.

Okay, well, I've run out of excuses and am unable to put it off anymore, so next time, I will finally reveal why I have been constantly using a constant acceleration of 10 (m/sec)/sec downward.

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