Calculus
posted by Antia on .
Two automobiles are travelling on intersecting roads. The first automobile is travelling northeast at 35 km/h. The second automobile begins 7 km north of the first, and it is travelling east at 25 km/h. Assuming that these cars will continue at this rate, will they collide? Assign vector equations to each line and assume that the first automobile begins at the origin.

Let (0,0) be where the first car started. Then its vector will be
u = 35/√2 t i + 35/√2 t j
= 24.75ti + 24.75tj
The second car starts at (0,7), so its vector is
v = 25ti + 7j
Is there a value for t where the two are equal? Let's see.
It will take car1 7/24.78=.2828 hr to travel 7km east and 7km north
It will take car2 7/25=.2800 hr to travel 7km east, staying at 7km north.
So, since .0028 hrs is about 10 seconds, there probably won't be a crash. 
Hmm. I see a typo. Car 1 travels at 24.7487, not 24.78km/hr
So, it takes 7/(35/√2) = √2/5 = 0.2828 hrs. Same answer.