Rate of change of separation distance

1. Oct 13, 2011

mishima

1. The problem statement, all variables and given/known data
This is from Calculus for the Practical Man, by J.E. Thompson, 1962 edition.

"Two automobiles are moving along straight level roads which cross at an angle of 60 degrees, one approaching the crossing at 25 mph, and the other leaving at 30 mph on the same side. How fast are they approaching or separating from each other at the moment when each is 10 miles from the crossing?"

2. Relevant equations
Definition of velocity. Basic differentiation operations.

3. The attempt at a solution
I start by finding what time it is since the leaving car left the crossing. At a distance of 10 miles, the time is 1/3 hr. This means the approaching car was at a distance of 10+25/3 miles when the leaving car was at the crossing. So I can calculate the average rate of change between the crossing and 10 mile positions to be 25 mph just by algebra but that's not what its asking. Also, just by thinking about it its obvious they are separating after the time of the equilateral triangle.

The answer given is that they are separating at 2.5 mph.

I'm sure the answer involves expressing the separation distance as a function of the other distances and then differentiating, but I can't come up with a geometrical rule that helps me. Its also kind of annoying that I can't tell if the angle between the roads should be 60 or 120, or if it is irrelevant in the final analysis.

2. Oct 14, 2011

mishima

Another thing is that the rate of change of separation distance is equal to zero when the car distances are 10 miles. This is because its at this instant that the rate of change of separation is flipping from negative (decreasing, approaching) to positive (increasing, separating).

3. Oct 14, 2011

mishima

Here's a picture if my wording was awkward. I'm trying to find a way to express S in terms of A and L. And remember, I can't use trig. Its probably something simple from geometry...

[PLAIN]http://img855.imageshack.us/img855/4172/calcpic.png [Broken]

Last edited by a moderator: May 5, 2017
4. Oct 14, 2011

HallsofIvy

Staff Emeritus
What I would do is take t= 0 as the moment that the two cars are 10 mi from the crossing. The time at which either car is at the crossing is irrelevant. At any time t, the distance from the crossing to the "leaving" car is 10+ 30t where t is in hours. Similarly the distance from the crossing to the approaching car is 10-25t. You can use the cosine law to find the distance between the two cars as a function of time, differentiate with respect to t to get a formula for the rate at which distance between the two cars is changing, and evaluate that at t= 0.

5. Oct 14, 2011

mishima

Thanks for responding, but one of the stipulations of the problem was that trigonometry isn't allowed. But yeah, cosine law is understandably applicable here.

Another thought was to use area somehow, but it doesn't seem to be the case that area is constant as 'S' slides around. Or project a parallelogram on the other parts of the crossing with 2A and 2L as diagonals. But then, I can't find any relation between a parallelograms sides and its diagonals.

6. Oct 15, 2011

mishima

Just went through the calculation, application of the cosine law indeed gives the correct answer. I guess then I need to find a way to express the cosine law in a purely geometrical way.

7. Jan 18, 2012

Achilleus30

I started working through all the problems in book too, and I had the exact same questions as you. I'm afraid I'm not much help. I'm just commiserating about the general ambiguous wording of some of the problems.

Also in the previous problem, problem #9, the book states "A three-mile wind blowing on a level is carrying a kite directly away from a boy. How high is the kite when it is directly over a point 100 feet away and he is paying out the string at the rate of 88 feet/min." One has to assume they mean a three mile per hour horizontal wind?

The ambiguous wording just made me want to switch to another book.