At t = 0 seconds
Two cars are travelling on perpendicular roads toward the same intersection. One of them, an Audi, is going west toward it at 52 km/hr. The other, a BMW, is going north toward it at 65 km/hr.
How far apart are they? No idea. We can't solve without some sort of fixed point. Thus, after THE SAME time period has elapsed: <== I made an assumption and I stated it.
t = r hours
the Audi is 500 m east of the intersection (already through!) and the BMW is 1.2 km south of the intersection (still approaching)
We may be expecting the vehicles to meet at the intersection. This appears not to be the case. We know that this distance is [math]\sqrt{500^2 + 1200^2} m[/math] We'll need this later.
How far are they from the intersection at any time after that?
t = s hrs = # of hours travelled after the first r hours that got us close to the intersection.
Audi: 500 m + s * 52 kmh = S(s)
Similarly
Beemer: 1200 m - s*65 kmh = Q(s)
Assuming 500m past and 1200m short are the same moment. <== I made an assumption and I stated it.
We're also assuming the speeds are constant. <== I made an assumption and I stated it.
Now, what is the distance between the two vehicles? Let's call it "D(t)". Then we have:
For just over one minute more, we have the Audi fleeing the intersection SLOWER than the BMW is approaching the intersection, sort of a chase. It is this moment that concerns us. At this moment, they are getting closer - the distance is DECREASING. We had better get a negative number by the time we think we're done.
[math]D^{2} = S^{2} + Q^{2}[/math]
Applying the derivative:
[math]D\dfrac{dD}{dt} = S\dfrac{dS}{dt} + Q\dfrac{dQ}{dt}[/math]
We already know DS/dt (52 kmh) and DQ/dt (-65 kmh) and D (1300 m) and S (500 m) and Q (1200 m). Substitute and solve for dD/dt.
Granted, it is a complicated problem and it required some assumptions. Welcome to the REAL world!! You just have to dig in and see where it leads. Don't give up because you cannot see the end from the beginning.
