How Do Related Rates Apply to Moving Objects in Differential Calculus?

[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?

 

[TaxOnFear]

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

 

[HallsofIvy]

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?

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.