Motion Questions. Perhaps the largest category of word problems concerns moving things and the related quantities distance, rate or speed and time. These are all related to one extraordinary formula, D equals RT. Some people like to remember this as the DRT equation. Thinking of that well you know, things move on the ground and the ground is made of dirt. Show Transcript
That helps some people to remember this formula. So in that formula it relates D distance, R which is rate or speed, and T which is time. Each one of these quantities has units, and it's important to keep track of the units in your calculations. The units for rate determine the units of both distance and time.
If rate is in miles per hour, distance must be in miles, and t must be in hours. If rate is in meters per second, distance must be in meters, and time must be in seconds. Similarly, if you know the units for distance and time, that determines the units for rate. The same length unit and time unit need to be used in the units of all three.
Notice that D equals RT is already solved for distance. And we can solve for rate or time if we need to. If we divide both sides of that initial equation by T, we solve for rate. So rate equals distance divided by time. Alternatively, go back to D equals RT. Divide both sides by rate, and what we get is T equals d over r.
Time equals distance divided by rate. That's solved for time. It's very important to be comfortable moving back and forth between any of those three forms. Let's do a few simple calculations for practice.
So pause the video, and then we'll talk about these three questions. Okay. The first question: What is the speed of someone who covers 240 miles in six hours at a constant speed? Well, we're given distance, we're given time, and we're looking for rate. So we want the version that is solve per rate.
Rate equals D over T. Then we're just gonna divide the distance by the time. 240, well 24 divided by 6 is 4, so 240 divided by 6 has to be 40. And that would be 40 and then miles over hours that's miles per hour. 40 miles per hour is the answer. Second question, how far does someone moving at 8 meters per second move in 40 seconds.
Here we're looking for the distance. So we want the form solve for distance, D equals RT, the original form, we're simply going to multiply the rate, 8 meters per second, times 40 seconds. Well 8 times 4 is 32, so 8 times 40 has to be 320. And meters per second times seconds gives us meters. So this should be 320 meters.
Finally, the third question, how much time does it take to move 300 feet at a speed of 20 feet per second? Well here we're looking for time, we know the distance and we know the rate,. So we need the form that is solve for time. Time equals distance over rate.
Now we're gonna divide that distance by that rate, of course we can cancel the 0s. That would leave us with 30 over 2, which would be 15. That's the number, and of course the time unit is second. So, time has to be in seconds. Has to be 15 seconds. Now each one of those is too easy, in and of itself, to be a complete test question, but these solutions demonstrate the fundamental skills you will need.
In other words, in any distance rate and time problem, you're going to have to typically be doing one of those calculations a few times. Okay, now I need to find this distance, okay, now I need to find that time. So you're gonna be putting together different rates and different times, and doing these calculations multiple times to get through a larger test question. What if the units given in the prompt and the units for which the prompt asks are inconsistent?
So, you might be given two different units in the prompt, and you might be given one unit in the prompt, and then ask for another unit. What do we do? Well, we need to change units, and we use a unit conversion, such as one hour equals 60 minutes, or one foot equals 12 inches. Those are two very standard unit conversions. Often the test will give you the conversions for any that aren't very common.
Any unit conversion can be written as an equation, but it can also be written as a fraction equal to one. So, we can say one equals one foot over 12 inches, or one equals 12 inches over one foot. You see, that's awfully useful. If we have something equal to one, that means we can always multiply or divide by that, and we don't change the fundamental values.
So, much in the same way, you can you could always multiply a fraction, say, by something like seven over seven. That doesn't change the value of the fraction because you're really multiplying by one. Much in the same way, you can take any quantity and multiply it by something like one foot over twelve inches, or twelve inches over one foot.
You could multiply by either of these, and it wouldn't change the value either, because this is also multiplying by one. So here's a question. Pause the video and then we'll talk about this. A lichen advances 4 cm each year across a rock slab. So four centimeters per year, that's the rate.
If this rate remains constant over time, how many years will it take to cross 30 meters? So here we're looking for time, so we want to solve for time. Time equals distance over rate. But we have to take care of the inconsistent units because we have a speed.
Involving centimeters, but a distance in meters. And of course the question is nice enough to tell us that one meter equals 100 centimeters. So our rate is 4 centimeters per year. Our distance is 30 meters. And the easiest thing to do is just right away change that distance to centimeters.
So we're just gonna multiply by 100. We're gonna multiply by 100 centimeters over 1 meter, cancel the meters. And we get 3000 centimeters. So that's the distance, and the distance and time are in the same unit. Time equals distance divided by rate. 3000 divided by four.
3000 divided by four is 750, and the time unit is year. So this is 750 years. Here's another question. Pause the video and then we'll talk about this. A car moving at 72 kilometers per hour moves how many meters in one second?
So we're told that 1 kilometer equals a thousand meters. Well, we also have to change hours to seconds. Well, we know that one hour is 60 minutes, and each minute has 60 seconds per minute. So 60 times 60 is 3600. One hour equals 3600 seconds. That's actually a very good number just to have memorized, cuz it does show up often enough that It's good that you don't have to recalculate it, just to know that one hour is 3,600 seconds.
Okay, well, now we can say that 72 kilometers over hours, 72 kilometers would be 72,000 meters, and one hour as we just thaw, saw, is 3600 seconds. So really, the speed is 72,000 meters over 3600 seconds. We'll cancel some zeroes. Now notice that 72 is just 2 times 36. So if 72 over 36 is 2, 720 over 36 has to be 20.
So 20 meters per second. So it moves 20 meters in one second. In summary. D equals RT is the fundamental equation for moving things. Know da, how to solve this equation for R or for T.
It's already solved for D. We wanna be able to solve it for each of the three variables. Keep track of the units of numbers when you plug in. Don't ignore units. If you pay attention to units, that will really help you. Unit conversions can be written as fractions equal to one.
We can multiply or divide by them. That's very convenient. Know the common unit conversions and change units, so that all units in the problem are consistent