Question

If two objects travel through space along two differentcurves, it’s often important to know whether they will collide.(Will a missile hit its moving target? Will two aircraftcollide?) The curves might intersect, but we need to knowwhether the objects are in the same position at the sametime. Suppose the trajectories of two particles are given bythe vector functions

\({r_1}(t) = \left\langle {{t^2},7t - 12,{t^2}} \right\rangle \) \({r_2}(t) = \left\langle {4t - 3,{t^2},5t - 6} \right\rangle \)

For \(t \ge 0\) . Do the particles collide?

Short answer

So the particles will collide at t = 3.

Step-by-step solution

Step 1: Rationalization

To check if the particles with trajectories

\({r_1}(t) = \left\langle {{t^2},7t - 12,{t^2}} \right\rangle \) \({r_2}(t) = \left\langle {4t - 3,{t^2},5t - 6} \right\rangle \)

We must find a value t that satisfies both vector equations in all components for \(t \ge 0\)If there does not exist a value t, that means the particles do not collide

Step 2: Equating the x component

By equating the x component we get

\(\begin{array}{l}{t^2} = 4t - 3\\{t^2} - 4t + 3 = 0\\(t - 3)(t - 1) = 0\\t = 1,3\end{array}\)

Now we have \(t = 1,3\) we can substitute these to the y components and z components and they must hold true for them to be a solution of the system. Also when \(t = 1\) then the x component is one whereas when \(t = 3\) the x component is nine.

Step 3: Equating the y component

By equating the y component we get

\(\begin{array}{l}when\;\;t = 1:\\{t^2} = 7t - 12\\1 = 7(1) - 12\\1 \ne - 5\end{array}\)

\(\begin{array}{l}when\;\;t = 3:\\{t^2} = 7t - 12\\9 = 7(3) - 12\\9 = 9\end{array}\)

Hence only \(t = 3\) remains the possible solution

Equating the z component

As t=1 cannot be the solution for both the x and y components then only t=3 is left to be checked for z component. For the z component we get:

\(\begin{array}{l}when\;\;t = 3:\\{t^2} = 5t - 6\\9 = 5(3) - 6\\9 = 9\end{array}\)

Step 5: Conclusion

Since there is a value which is t=3 that satisfies both equations, the particles will collide and this will occur at t=3 at the position \((9,9,9)\).

So the particles will collide at t = 3.