A 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? Choose 1 answer: (A) S(t)=(k)/(t^(2)) (B) S(t)=(k)/(S^(2)) (C) (dS)/(dt)=(k)/(t^(2)) (D) (dS)/(dt)=(k)/(S^(2))

Question

A 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? Choose 1 answer: (A)  S(t)=(k)/(t^(2)) (B)  S(t)=(k)/(S^(2)) (C)  (dS)/(dt)=(k)/(t^(2)) (D)  (dS)/(dt)=(k)/(S^(2))

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Understand the problem: Understand the problem. We are given that the speed of a particle is inversely proportional to the square of the distance it has traveled. This means that as the distance increases, the speed decreases in such a way that the product of the speed and the square of the distance remains constant. We need to find the equation that correctly represents this relationship.Translate into equation: Translate the given information into a mathematical equation. If S represents the distance traveled and v represents the speed, then we can say that v is inversely proportional to S^2. This can be written as: v = k / S^2 where k is a constant of proportionality.Determine correct form: Determine the correct form of the equation. We are looking for an equation that describes the relationship between speed and distance. Since speed is the derivative of distance with respect to time, we can write the equation as: (dS/dt) = k / S^2 This equation shows that the rate of change of distance with respect to time (which is speed) is inversely proportional to the square of the distance traveled.Match with options: Match the equation with the given options. The equation we derived in Step 3 is (dS/dt) = k / S^2. This corresponds to option (D) in the given choices.

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