How can I know whether the airplanes collide by using parametric equations

Question

Recall that a line hes equation y=mx+c. Suppose one airplane moves along the line y=2x+3 while the other airplane moves along the line y=3x-2. By plotting a graph, even though the lines are intersect, but the equations did not tell us whether there will be a mid air collision. I know that the concept of parametric equation can be used to explain this. Can anybody show me an example?

Answer 1

The paths might intersect, but the planes might not be at that point at the same time. Just telling us where the paths go (which is what an equation like $y = mx + c$ does) doesn't help in clearing this up.

However, with parametrized lines, you get the additional information of "where was the plane at a specific time". For instance, if plane $1$ follows the parametrization $$ \cases{x_1(t) = t\\ y_1(t) = mt + c} $$ which would trace out exactly the same line, then we can tell that after $2$ seconds, the plane is at the point $(2, 2m+c)$. We can also reverse this, and ask "At what time was the first plane at the intersection point?" The immediate follow-up would then be "What about the second plane? Did they collide?"

Note that if you have two parametrized lines, and you want to find the point of intersection (regardless of time), then you need to use $t$ to parametrize one line and $s$ to parametrize the other. This is again because the planes might have been at the intersection point at different times. If you use the same parametrization variable for both planes, and try to solve, you're implicitly asking "At what time were the planes at the same point in space?" which might not have happened.

However, if you use two different parametrization variables, you're asking "Are there two times $t_0$ and $s_0$ such that the place where plane $1$ was at time $t_0$ and plane $2$ was at time $s_0$ is the same place?" You can see that this is really asking for where their paths intersect, not caring about whether they actually collided. Of course, afterward you can go in and check whether $s_0 = t_0$, and if that's true then they did collide.

Answer 2

Notice, we have the paths of airplanes along the following straight lines $$y=2x+3\tag 1$$ & $$y=3x-2\tag 2$$ On solving (1) & (3), we get intersection point of the straight lines $(5, 13)$

Now, from the given equations of the straight lines paths, we can conclude that the mid-air collision of airplanes can take place only at the point $(5, 13)$ $$\bbox[5pt, border:2.5pt solid #FF0000]{\color{blue}{\text{Point of the collision of airplanes is }\ (5, 13)}}$$

Hence, we have exact idea of the point/location of collision where it can take place. Because it can take place only at the point of intersection.

But we do not have any idea about the time of collision when it takes place. Because time of collision depends on the initial locations of the airplanes & their respective speeds.

So we are sure about the point/location of collision not about the time.
However to know the $\color{red}{\text{time of collision}}$, we need to know their initial locations w.r.t. point of collision $(5, 13)$ & respective speeds.

In fact, the collision will take place if and only if both the planes reach at the point $(5, 13)$ at the same time.