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The Concept of Related Rates

  1. Dec 3, 2011 #1
    Hi Guys,

    I have a general question (not necessarily a homework question) about the concept of related rates in differential calculus.

    Most related rates problems present to you a question that generally asks,

    After x time has elapsed,
    At t= __,

    what is the rate of change between __ and __?

    Suppose I have a related rates problems involving kinematics.

    At noon, object A is __units away from object B. Object A is moving in an opposite direction away from object B. Object A is moving at a constant rate of __units and object B is moving at a constant rate of __units.

    After 4 hours, what is the rate of change of the distance between object A and B.

    I am content with the premise. What captures my interest is the question.

    If two objects are both moving opposite one another at a constant speed, wouldn't the distance between them be changing at a constant speed as well? Why would the derivative of the distance between the objects with respect to time be different after any time?
  2. jcsd
  3. Dec 3, 2011 #2
    When I see rate in kinematics, I automatically think of acceleration. Different accelerations, different derivitives.
  4. Dec 3, 2011 #3


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    Science Advisor

    Yes, it would. However, one thing that is missing is the statement what the speeds are measure relative to. If you are saying that A is moving with velocity v relative to B then B must be moving with velocity -v relative to A. On the other hand if you are saying that B is moving with speed v1 relative to some third point, C, and that A is moving at speed v2 relative to C, then A and B are moving (ignoring relativity!) at speed v1+ v2 relative to each other.

    The way you state it, "moving opposite one another", that's not a very interesting question for exactly the reason you state- the relative speed is constant.

    A more interesting question would be "If B is moving with speed v1 due east relative to C and A is moving with speed v2 due west relative to C, how fast is A moving relative to C?"

    Now, the distance between A and C is given by a quadratic equation (the Pythagorean theorem) and the relative speed is not constant.
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