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FUN‑7 (EU)

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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 so 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 so 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 wrote 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 up what we have up here now let's see let's see what 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 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 D s 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 it has traveled so we like that choice