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Average speed for entire journey - solved numerical

Let's learn how to calculate average speed when someone covers equal distances in two different speeds. Created by Mahesh Shenoy.

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Video transcript

[Teacher]- A biker boy decides to have an adventure by climbing this mountain on his bike. So, he starts his engine, and starts climbing up at a speed of one kilometer per hour. He eventually reaches the top, and without wasting any time, quickly decides to come back down. Now, while coming back, it turns out he has a speed of nine kilometers per hour, much faster. And eventually, he reaches the bottom, at the exact same spot where he started. And the question that we have to answer in this video is what was the biker boy's average speed for the entire journey? Oh, boy! This brings back a lot of memories because I use to struggle to solve this problem. So this is going to be super exciting. Alright! Let's go ahead and do this! Okay, so let's go ahead and solve this. Let's start by recalling what is the formula for average speed. Average speed is basically how much distance something is traveling in a unit of time. And so it's going to be distance divided by time. Which means in this problem, I need to take the total distance traveled by this biker boy to go up and come back down, and divide it by the total time it took for him to complete his journey. And now you can see what the problem is. I don't know either of them. Distance is not mentioned, nor is the time mentioned. All that is given to us are the speeds of the boy when he's going up and when he's coming back down. So how do we solve this? So for example, while going up we know he's traveling one kilometer per hour. This does not mean that this is one kilometer and it takes him one hour. This does not mean that. It could mean that this is ten kilometers long, and it took him ten hours, right? That also means one kilometer per hour. It could also mean a hundred kilometers and a hundred hours. You see what I mean? Distance is not given, and times are also not given. How do we calculate? This is where thought, when I was solving this for the first time, maybe this is a trick question. Maybe there's another way to solve this. What I'm thinking is I have two speeds given to me; average speed while going up and speed while coming back down. And I'm asked, what is the average? Well, maybe we can just go ahead and average them like how we usually do in maths. So, I thought maybe we can just add them and divide by two. Why not? And so that would be one plus nine, that is ten, divided by two. And that's going to be five kilometers per hour. That is the average value, isn't it? I have the answer. And I used to feel super proud because I used my intellectual brain. But, guess what? This is wrong. And I would be like, what? Why is this wrong? I mean, I understand that I'm not using the formula, but, hey, this is how we do it in maths, right? So, why can't I do it this way? And so, you know what? I think it would be a good idea to first spend a couple of minutes, maybe even more, to try and understand why this method is wrong so that, you know, we don't make mistakes like this in the future. And then look at what is the right way to do this. So why is this method wrong? Why? Why? Why? Well, in short, this method assumes that the time it took for the bike to go up and come back down is exactly the same. In other words, this answer is only correct provided the up time is exactly equal to the down time. So let's see how. There are multiple ways to see why this means up time is the same as down time. But one way, which I like to do it, is by taking examples. So, let's assume that it took him one hour to go up, and one hour to come back down. That's not correct. That's not true, but let's assume that. Then what would be the average speed? Well, if it took him one hour to go up, he would travel one kilometer up, and similarly nine kilometers down. Then the total distance he would have traveled would be one plus nine. And what would be the total time it takes him? Well, one hour and one hour- the total time would be two hours. So if we divide, what will be the average speed? The average speed would be ten divided by two which is exactly five kilometers per hour. So do you see that this answer only works when the time it takes to go up is exactly the same as it takes to come back down? And just to be clear, it doesn't have to be one hour and one hour. That was just an example. Even if it took him two hours to go up and two hours to come back down, the answer would be the same. Let's see how. If he goes up for two hours, he will travel twice the distance. If he comes back down for two hours, again he will travel twice the distance. So, the total distance will be twice. Makes sense, right? Because he's going for twice the climb now. And similarly, the total time taken will also be twice. Two hours and two hours will be four hours. And notice when you do the average, again this cancels and we end up with the same answer. So, regardless of how much time was the actual time, regardless of that, as long as his up time is equal to his down time, then this method works, and the answer will be exactly five. Right in between one and nine. But in our example, do you think the up time is the same as the down time? Do you think it takes him the same time to go up as it takes him to come back down? The answer is no. He doesn't. Because while going up, he travels super slow. So, if you could see it, if you could see the animation, notice it's gonna take him a long time to travel upwards. Yep, this is gonna take a while. There it is. And once he reverses and comes back down he travels super fast, so it takes him a very small time to come back down. So, the up time is not the same as the down time. It takes him much longer to travel up compared to traveling down. And as a result, the answer cannot be five kilometers per hour. That's why this method is wrong. So now that we know that this is not the right answer, here's another question before we go ahead and solve it. Do you think the actual answer is going to be less than five? Or more than five? Please give this a thought so that our concept will be very, very clear. So pause the video and think about this for a while. Okay, now if you got less than five, then you are absolutely correct. Why, you may ask? Well, here's how I like to think about this; we saw that if the biker boy had traveled for equal times up and down, then the answer would be five, right? Which is exactly between one and nine. Five is exactly between one and nine. But we saw that in reality, he spends much longer being at one kilometer per hour. So since he spends more time being at one kilometer per hour, can you kind of see that the average value should shift closer to one? Which means the answer should be somewhere between one and five. And as a result, our average speed should be less than five kilometers per hour. In fact, it's very easy to see this if we exaggerate. Let's say he traveled upwards for a thousand hours. And downward only for one hour. Now we know that's not going to be true, but let's see what the answer would be. If he traveled up for one thousand hours, the distance traveled would be a thousand kilometers up, nine kilometers down. So the total distance would be one thousand and nine kilometers. And what would be the total time taken? A thousand hours up, one hour down. One thousand and one hours. So the average speed would be- look at this number. This number is almost one thousand by one thousand. This is very close to one, isn't it? One kilometer per hour. It's a little bit larger than one, if you see properly. But it's very close to one. Can you see that? So hopefully this helps us understand the average speed should be closer to one, because he spends more time being at one kilometer per hour. So, now that we've understood what not to do, let's go ahead and solve this. So, let's get rid of all that stuff and see if we can use this formula. This actually circles back to our original question. How do we use the formula? Because distance is not given. And even time is not given. Well, we can always assume, right? So, let's assume that this distance that our biker boy travels is d. Then what's the total distance that he travels? Well, that's going to be d while going up and d while coming back down. So the total distance is 2d. What about time? Well, we've already seen that the time taken is not the same. There's the whole point over here. While going up, we saw the time taken was more. Let's call that time t1. And while coming back down, let's say that time is t2. Then what's the total time? That will be t1 plus t2. Alright, what next? What do we do after this? Can we somehow figure out what t1 and t2 are? Let's see. If you look at the upward motion, we know the distance traveled is d. We know the speed. Oh! That means we can calculate the time t1 is. Speed equals distance by time, so we can write what t1 is in terms of d and speed. Similarly, for the downward motion we can do the same thing. We know the distance is d. We know the speed. That means we can write what t2 is. So this means we can write what t1 and t2 are in terms of d, and then maybe we can substitute it over here, and see if we can simplify. So, why don't you take some time to see if you can try this yourself first. And then we'll do it together. Alright, let's start with the upward motion. Let's use speed equals distance over time. Speed is one kilometer per hour. I'll not write the units. Equals distance, that is d, over time. And time is t1. This means time (t1) is going to be distance divided by speed. So, t1 is going to be d over one, and d over one is just d in this case. So I'm just going to write it as d. Let's do the same thing for the downward motion. For downward motion again we'll use speed equals distance over time. Speed for downward motion is nine kilometers per hour. So let's write nine equals distance divided by time (t2). And again now t2 becomes d over nine. And now we can plug these things over here, and see if we can get the answer. Alright, let's do this. So, if you plug in average speed it's going 2d divided by t1, which is just d plus t2, which is d over nine. And all we have to do now is the algebra. So let's see what we get. We have 2d on the numerator. In the denominator, d is common. So let's pull that d out. And the reason I'm pulling it out, you can see, if I pull d out, I'll get one plus one over nine. And the reason I'm pulling that d out is because the d is going to get canceled out. And that means we're going to end up with a number. That's good news. Alright, so what's that going to be equal to? Well, we have two in the numerator divided by- let's take the common denominator here. If you take the common denominator and multiply this by nine- let's see. Let me shift this up a little bit. So the common denominator is nine. I have to multiply this by nine plus one. That makes it two over ten divided by nine. I have to be a little bit careful because I have a fraction in the denominator. What I like to do is convert that division into multiplication. I'll do two into the reciprocal of the denominator. That gives us 18 over 10. That is one point eight. One point eight kilometers per hour. And that is our answer. So let's put all of this in one fame now. Okay, so there it is. This is our average speed. The average speed for the entire journey is one point eight kilometers per hour. And guess what? We predicted this, right? We predicted that the answer has to be less than five kilometers per hour. It has to be closer to one because he spends more time traveling at one kilometer per hour. Hopefully all of the now makes sense. So what did we learn in this video? We learned what not to do. The most common mistake I used to make was add and divide by two. Hopefully now we know why not to do that. And the best way to solve any problem involving average speed is usually to just go by the formula and try and solve it.