A cyclist traveled 36.5 kilometers from point A to point B at 36.5 kilometers per hour and then traveled a further 22 kilometers in the same direction at 22 kilometers per hour. Given that 𝐧 is a unit vector in AB direction, determine the cyclist’s average speed 𝑣 and average velocity vector 𝐯 during the whole trip.
Now, we’re given two speeds for two different parts of the journey and then asked to calculate the average speed for the entire trip. We need to be careful. In general, we can’t just find the average of the two speeds given. What we actually have to do instead is begin by finding the total distance traveled and the total time it took. Then we use the formula average speed equals distance divided by time to find that average speed 𝑣. So let’s begin by finding the total distance traveled.
During the first part of the journey, the cyclist traveled 36.5 kilometers at a speed of 36.5 kilometers per hour, so we could rearrange the speed–distance–time formula. However, if we think about this logically, if he travels 36.5 kilometers per hour and travels for 36.5 kilometers, then it simply must take exactly one hour. And so the journey from point A to point B is 36.5 kilometers, and it takes one hour. We can actually repeat this process for the second part of the journey. The cyclist travels another 22 kilometers at a speed of 22 kilometers per hour. And so this part of the journey must also take exactly one hour. So the second part of the journey, the part for B onwards, is 22 kilometers and it takes one hour.
We find the total distance traveled by adding 36.5 and 22, and we get 58.5 kilometers. Similarly, we can add one hour and one hour and we find that the total journey takes two hours. And so the average speed 𝑣 is the total distance traveled divided by the total time. That’s 58.5 divided by two, which is 29.25 kilometers per hour. Now, you might notice that 29.25 is simply the average of the two speeds, the average of 36.5 and 22. But at the beginning of this question, we said that we couldn’t generally just answer these sorts of questions by finding the average of the two speeds. It does, however, work when the two part of the journeys are the exact same time. So since the journey from A to B was one hour and the journey for B onwards was one hour, we could have used this method.
Now that we know the average speed 𝑣, let’s consider how we’re going to find the average velocity vector 𝐯. Now, if we think about this pictorially, we know the journey from A to B will look a little something like this. 𝐧 is a unit vector in the AB direction, and we know that the relationship between speed and velocity is that speed is essentially the length or the magnitude of the velocity vector. Since 𝐧 is a unit vector, we can therefore say that the velocity must be 29.25 multiples of that unit vector. In other words, the average velocity vector 𝐯 during the whole trip is 29.25𝐧 kilometers per hour.