Video Transcript
A fish swims a distance of 15 meters to the left at an average speed of two meters per second and then immediately turns around and swims half this distance to the right. The time that passes while all this occurs is 12 seconds, as shown in the diagram. In this question, consider displacement to the left as positive. What is the displacement of the fish from its starting point 12 seconds after it starts moving?
Then, there are three further parts to this question, which we’ll consider in turn. But to answer the first part, we need to define some of the key terms. Here, we want the displacement of the fish. And the question tells us that the displacement to the left is going to be positive. So what does this mean, to have positive displacement? Well, we recall that displacement is a vector quantity. It has both direction and magnitude. The magnitude of the displacement is known as distance. So with this in mind, how do we find the displacement of the fish from its starting point 12 seconds after it starts moving?
Well, we’re told that it swims to the left 15 meters, turns around, and swims half this distance to the right. This is the point at which the fish is located after 12 seconds. So we might begin by calculating the value of 𝑑 over two. And we’re told that 𝑑 is 15 meters. This is the total distance that the fish swam to the left. This means 𝑑 over two is 15 over two or 7.5 meters. This means that the fish then swims 15 meters to the left, turns around, and swims seven and half meters to the right. The displacement of the fish at this point from the starting point will be the distance between the starting point and this location.
We might consider this as sort of a one-dimensional vector. The fish has a displacement of 15 meters plus negative 7.5. We add negative 7.5 because the fish swims to the right. That’s the negative direction. 15 add negative 7.5 is 7.5. So the displacement of the fish from the starting point 12 seconds after it starts moving is 7.5 meters. We’re going to call this displacement 𝑠. And then we’re going to add this in the corner and clear some space for the next question.
The second part of this question asks us, for how long does the fish swim to the left?
And so to answer this, we’re going to go right back to the start. We’re told the fish swims 15 meters to the left at an average speed of two meters per second. Since this is in a single direction, we can consider these as being magnitudes. That’s why we’re working with distance and speed rather than displacement and velocity. And so we can calculate the time that the fish swam to the left in by using the formula speed equals distance over time. Defining this time to be equal to 𝑡, we can write this as two equals 15 over 𝑡. To solve for 𝑡, we’ll multiply by 𝑡 and then divide by two. So 𝑡 equals 15 divided by two, which equals 7.5 or 7.5 seconds. Now, specifically, since that represents the first part of the journey, let’s call that 𝑡 sub one and clear some space for the third part of this question.
The third part of this question says, what is the fish’s average velocity while it moves to the right? Answer to one decimal place.
Now this is average velocity, not average speed. We know that velocity, like displacement, is a vector quantity. It has magnitude and direction. In this case, the velocity of the fish if it’s moving to the left will be positive and if it’s moving to the right will be negative. This is a direct result of the formula that’s a little bit like the speed–distance–time formula. This time, though, its velocity equals displacement over time. Since the fish moving to the right has a negative displacement, this means we’re going to be dividing a negative by a positive; that’s time. That will give us a negative result.
Now, we know that when the fish is traveling to the right, it travels a distance of 7.5 meters. But this is in the opposite direction to the positive direction, so the displacement is negative 7.5. The time that it takes is the difference between the total time, 12 seconds, and the time it takes to move to the left, which we calculated to be 7.5 seconds. So the time is 12 minus 7.5. That’s 4.5. So velocity is negative 7.5 over 4.5, and that’s negative 1.666 or negative 1.6 recurring. Correct to the nearest 10th or one decimal place, that’s negative 1.7 meters per second. So that’s the fish’s average velocity while it moves to the right. Let’s clear some space and answer the fourth and final part of the question.
The fourth part of this question says, what is the average velocity of the fish over the 12 seconds that the fish moves for?
We’re still going to use the same formula: velocity equals displacement over time. But of course, we’re interested in the velocity of the fish over the entire 12 seconds. So that’s going to be the time we’re going to use. And remember, we calculated the total displacement of the fish after 12 seconds. That was 7.5 meters. Since the displacement is positive, our velocity is also going to be positive. It’s 7.5 divided by 12, and that’s 0.625. And that’s our final answer. The average velocity of the fish over the whole 12 seconds is 0.625 meters per second.
