Suppose the trajectories of two particles are given by the vector functions

Question

Suppose the trajectories of two particles are given by the vector functions

r₁(t) = <t², 7t-12, t²>

r₂(t) = <4t-3, t², 5t-6>

for t > 0. Do the particles collide? If so, find the point of intersection(s).

Answer 1

Hi! So the basic idea of this problem is simple: Two objects collide if all three components of their position vectors are the same at any point in time.

In other words, this means they collide if their x, y, and z coordinates are the same at some time.

So we want:

t² from r₁(t) to equal 4t-3 from r₂(t),

7t-12 from r₁(t) to equal t² from r₂(t), and

t² from r₁(t) to equal 5t-6 from r₂(t)

In math equations, we must solve:

t² = 4t - 3,

7t-12 = t²,

t² = 5t-6

You can factor or use the quadratic formula to solve all three of these. You should find that all three have a similar answer of t = 3. This means that at t = 3 all three equalities are true, so the objects will collide.

To find the point of intersection, plug in t= 3 into either r₁ or r₂ and you'll find their positions.