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.
To determine if the two automobiles will collide, we need to find the point where their paths intersect.
Let's assign a vector equation to each line:
For the first automobile:
r1(t) = <0, 0> + t<cos(45°), sin(45°)> = <t*cos(45°), t*sin(45°)>
For the second automobile:
r2(t) = <0, 7> + t<cos(90°), sin(90°)> = <0, 7> + t<1, 0> = <t, 7>
Here, t represents time in hours.
Now, we need to find the time when the x and y coordinates of both automobiles are the same.
For the x-coordinate:
t*cos(45°) = t
cos(45°) = 1
This equation is always true, so the x-coordinate of both automobiles will always be the same.
For the y-coordinate:
t*sin(45°) = 7
sin(45°) = 7/t
1/√2 = 7/t
t = 7√2
t ≈ 9.9
So, the two automobiles will collide approximately 9.9 hours after the first automobile starts moving.
Please note that this is a simplified model and assumes both automobiles continue moving at a constant speed. Real-world factors such as traffic, road conditions, and the actions of the drivers can influence whether a collision occurs.