Two particles are traveling through space. At time t the first particle is at the point (-1+t, 4

step 1 We have two particles here and their paths are described as such. I'm going to rewrite these pat

Two particles are traveling through space. At time t the...

Key Concepts

Parametric Equations of Motion

This concept involves representing the trajectory of a particle using equations that express each coordinate (x, y, z) as a function of a parameter, typically time. This allows us to describe the motion of objects in space, capturing both the direction and speed along a continuous path.

Collision versus Intersection

Collision refers to the event where two particles are at the same position at the same time, meaning both their spatial coordinates and time parameters match. Intersection of paths, on the other hand, means that the spatial paths of the two trajectories cross at a common point, regardless of whether the particles are there simultaneously.

Intersection of Paths in Space

This involves determining the point in space where two trajectories meet. It requires finding a common solution to the parametric equations of both paths. Even if the particles do not collide (i.e., they do not reach that point at the same time), the paths can still cross at a point which is the geometric intersection of their trajectories.

System of Equations in Parametric Context

When analyzing problems involving particle motion, solving for possible collisions or intersections typically leads to a set of simultaneous equations derived from equating the parametric expressions. This system of equations must be solved to check if and when both conditions—similar positions and (for collisions) matching time parameters—are satisfied.

Transcript

00:01 We have two particles here and their paths are described as such.

00:04 I'm going to rewrite these paths in a separated form.

00:10 So particle one, its position as a function of t is negative one, four, negative one, plus, let's see, one minus one, two, times t.

00:24 And particle two is negative seven, negative six, negative one, plus two, two, one, times t.

00:35 What i've done here is just take the constant parts and put those in a vector, and then the t parts and put those in another vector.

00:45 So now we can describe the two paths in words.

00:47 This is a path, p1, is a path that starts at negative one, four, negative one, and then continues in a linear fashion like that, in a direction parallel to two, one, negative one, two.

01:06 Likewise, path two starts at the point negative seven, negative six, negative one, and continues on in a line parallel to the vector two, two, one.

01:17 That's the path in words.

01:21 Do the two particles collide? well, let's see.

01:24 That would require there to be a value t, such that p1 of t is equal to p2 of, excuse me, not of s, of t, because they need to be at the same place at the same time.

01:36 So we can just set the components equal to each other.

01:42 Negative one plus t should be equal to negative seven plus two t.

01:49 Okay, well, this is one equation with one unknown, so we can go ahead and solve this real quick for t.

01:54 I'm subtracting t, negative one equals negative seven plus t, add seven, six equals t.

02:03 There we go.

02:03 So if they collide, it will have to be at time t equals six.

02:07 Does this work out in general? well, let's see.

02:13 P1 of six is equal to, excuse me, negative one plus six is five.

02:22 Four minus six is negative two, and then negative one plus 12 is 11.

02:30 P2 of six is equal to, okay, negative seven plus 12 is five.

02:35 That's exactly what we want.

02:37 Negative six plus 12 is six.