Shortest distance between two cars

[Adjoint]

Homework Statement

Two straight roads, which are perpendicular to each other, cross at point O.

Suppose a car is at distance 250m from the origin on one road, and another car is at distance 350m from the origin on another road.

Both cars are approaching towards the origin.

The first car has a constant velocity of 6m/s and the second car has constant velocity of 12m/s.

When does the distance between the two cars become shortest? And what's that shortest distance?

Homework Equations

The Attempt at a Solution

Lets suppose at time t the cars' distance becomes shortest.
So at that time the first car's position will be (0, 250 - 6t) and the second car's position would be (350 - 12t, 0)

So distance between them is √{(350 - 12t)² + (250 - 6t)²}

Next suppose A = (350 - 12t)² + (250 - 6t)²
For minimum dA/dt = 0 from here I get t

Is my approach ok? (I am not much expert in calculus.)

 

[Orodruin]

Looks reasonable. Solve for t in dA/dt = 0 and insert into your expression for the distance.

 

[Adjoint]

Thanks.

A = 180t² - 11400t + 185000
dA/dt = 360t - 11400 = 0 gives t = 31.7
And the minimum distance is √{(350 - 12t)² + (250 - 6t)²} = 67.1

 

[rude man]

Agreed.

 

[HalfLostAndSearching]

I would say that your approach is a good start. You have correctly identified the positions of the two cars at any given time and have correctly set up the equation for the distance between them. Your next step of finding the minimum value of this distance using calculus is also a valid approach.

However, I would suggest considering the physical implications of this problem as well. Since the two cars are approaching each other, the shortest distance between them will occur when they are both at the same point, i.e. when their positions are equal. This means that the time at which the distance between them is shortest can be found by setting the two positions equal to each other and solving for t.

Additionally, it might be helpful to plot the positions of the two cars on a graph to visualize the situation and better understand the relationship between the two variables (time and distance). This can also help in finding the shortest distance between the two cars.

Overall, your approach is on the right track and with some additional considerations, you should be able to find the solution to this problem.