A vertically upward-pointing ladder has rungs that are spaced 15 centimeters apart, with the first rung 15 centimeters above the ground. A construction worker climbs eight rungs up the ladder and then climbs down two rungs. How many centimeters above the ground are the worker’s feet displaced?
Let’s start by underlining the important information in this question. Well, we know that the ladder is vertically upward pointing. We also know that it has rungs spaced 15 centimeters apart. Thirdly, we know that the first rung is 15 centimeters above the ground. That’s all the important information about the ladder. But, there’s more in this question. A construction worker goes up and down that ladder. What do they do? Well, they climb eight rungs up the ladder and then they climb down two. What this question wants us to find out is how many centimeters above the ground the worker’s feet were displaced.
Let’s start this off by drawing a diagram. So this is how we’ve drawn the ladder with the first rung 15 centimeters off the ground and each subsequent rung 15 centimeters away from the previous one. Now, here is an interesting point. Since we know that the ladder is vertically upward pointing, that’s what it says at the beginning of the question, we can afford to draw this diagram as we have. The ladder is not resting on a wall or leaning over or anything. It’s pointing straight up. That means that we don’t need to worry about what the ladder is doing in any other direction apart from up or down. So we can draw it like this face on, as if we’re about to climb the ladder.
So with that being sorted, let’s look at what the construction worker does. Well, we know that they climbed up the ladder one rung at a time until they climbed eight rungs. That’s the first step. But then immediately after, they climbed back down two rungs. Now it’s really important to be careful here. The question is not asking us for the total distance travelled by the construction worker. Instead, it’s asking us for the displacement. Displacement is defined as a vector whose magnitude or size is equal to the shortest distance between the start point and the finish point. So keeping that in mind, we know that the starting point of the construction worker was down here at the bottom of the ladder. And the finish point was up here, after they climbed up eight rungs and down two rungs. That distance is the same as just climbing up eight minus two which is six rungs.
As a very quick aside by the way, it’s important to know that displacement is defined as a vector. Vectors have both, size or magnitude and direction. However, in this question, it’s kind of irrelevant. We don’t need to know the direction because we know that the construction worker can only climb up the ladder. What we’re more interested in is the distance between the start point and the finish point. And that is equivalent to climbing up six rungs of the ladder. Now since each rung is 15 centimeters away from the previous one and the first one is 15 centimeters away from the ground, we can work out that the distance between the start point and the finish point is 15 centimeters times six rungs. That ends up being 90 centimeters.
Another way to think about this is to not worry about the number of rungs. Instead, we can just think about the distance the construction worker has gone up or down the ladder. When they first went up eight rungs, that was the same as going up 15 centimeters times eight which was 120 centimeters. Then when they came back down two rungs, that was the same as going down 15 centimeters times two which is 30 centimeters. So the total displacement was this distance here. And that distance is actually 120 centimeters minus 30 centimeters. And that ends up being, surprise, surprise, 90 centimeters.