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) (dS)/(dt)=(k)/(t^(2)) (B) S(t)=(k)/(t^(2)) (c) (dS)/(dt)=(k)/(S^(2)) (D) S(t)=(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)  (dS)/(dt)=(k)/(t^(2)) (B)  S(t)=(k)/(t^(2)) (c)  (dS)/(dt)=(k)/(S^(2)) (D)  S(t)=(k)/(S^(2))

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Identify Relationship: Identify the relationship between speed and distance. Since the speed is inversely proportional to the square of the distance, we can write this relationship as: Speed = k / S^2, where k is a constant of proportionality.Translate into Equation: Translate the relationship into a differential equation. The speed of the particle is the derivative of the distance with respect to time, which can be written as dS/dt. Therefore, we can express the relationship as: dS/dt = k / S^2Match to Options: Match the relationship to the given options. We need to find the option that correctly represents the relationship dS/dt = k / S^2. Let's compare this to the given options: (A) (dS)/(dt)=(k)/(t^(2)) - This option suggests that speed is inversely proportional to the square of time, which is not the relationship we have. (B) S(t)=(k)/(t^(2)) - This option suggests that the distance itself is inversely proportional to the square of time, which is not the relationship we have. (C) (dS)/(dt)=(k)/(S^(2)) - This option matches our relationship exactly. (D) S(t)=(k)/(S^(2)) - This option is not meaningful, as it suggests that distance is inversely proportional to the square of itself.

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