# The Concept of Related Rates

1. Dec 3, 2011

### nDever

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,
or
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. Dec 3, 2011

### TaxOnFear

When I see rate in kinematics, I automatically think of acceleration. Different accelerations, different derivitives.

3. Dec 3, 2011

### HallsofIvy

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.