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Writing a differential equation | Differential equations | AP Calculus AB | Khan Academy

2m read
·Nov 11, 2024

Particle moves along a straight line. Its speed is inversely proportional to the square of the distance s it has traveled. Which equation describes this relationship?

So I'm not going to even look at these choices and I'm just going to try to parse this sentence up here and see if we can come up with an equation.

They tell us its speed is inversely proportional to what? To the square of the distance s it has traveled. So s is equal to distance. S is equal to distance. And how would we denote speed then if s is distance? Well, speed is the rate of change of distance with respect to time.

So our speed would be the rate of distance with respect to time, the rate of change of distance with respect to time. So this is going to be our speed.

Now that we got our notation, the S is the distance, the derivative of s with respect to time is speed. We can say the speed, which is D capital S DT, is inversely proportional.

So it's inversely proportional. I WR a proportionality constant over what? It's inversely proportional to what? To the square of the distance, to the square of the distance it has traveled.

So there you go. This is an equation that I think is describing a differential equation really that's describing what we have up here. Now, let's see which of these choices match that.

Well, actually this one is exactly what we wrote. The speed, the rate of change of distance with respect to time, is inversely proportional to the square of the distance.

Now just to make sure we understand these other ones, let's just interpret them. This is saying that the distance, which is a function of time, is inversely proportional to the time squared. That's not what they told us.

This is saying that the distance is inversely proportional to the distance squared. That one is especially strange.

And this is saying that the distance with respect to time, the change in distance with respect to time, the derivative of the distance with respect to time ds/dt or the speed, is inversely proportional to time squared. Well, that's not what they said. They said it's inversely proportional to the square of the distance it has traveled.

So we like that choice.

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