Question

If two objects travel through space along two different curves, it’s often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of two particles are given by the vector functions: r1(t)= r2(t)=<4t-3, t^2, 5t-6> for t>=0.Do the particles collide?

Answer 1

Answer:

Both objects collide at t=3 in the point  <9,9,9>

Step-by-step explanation:

Collision Of Moving Objects

Two objects can describe different trajectories in the space. Those trajectories can intersect in one or more points but it doesn't mean they collide. Collision occurs if they are in the same position at the same time. If we know the positions as a function of time of each object, we could try so find if, for a given time, they are in the same position.

The positions of two object are given as

$r1(t)=$

$r2(t)=$

Let's find out if there is at least one value of t that makes both positions to be the same. We can try by equating one of the three coordinates and testing if the value of t make both have the same x,y,z coordinate. Let's try equating the x-components of both

$t^2=4t-3$

Rearranging

$t^2-4t+3=0$

Factoring

$(t-1)(t-3)=0$

We found two solutions

$t=1,\ t=3$

for t=1 the x-coordinates are

$x1=t^2=1$

$x2=4t-3=1$

For t=3

$x1=t^2=9$

$x2=4t-3=9$

Now we'll test both values in the y-coordinates

$y1=7t-12$

$y2=t^2$

For t=1

$y1=-5$

$y2=1$

Thus they don't collide at t=1. Let's try t=3

$y1=7(3)-12=9$

$y2=3^2=9$

Now let's try the z-coordinate for t=3

$z1=t^2=9$

$z2=5t-6=9$

Since the three coordinates match, we can say both objects collide at t=3 in the point  <9,9,9>