Problem 67 - Essential Calculus Early Transcendentals

Question

If two objects travel through space along two different curves, it's often important to know whether they will col- lide. (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 $$\mathbf{r}_{1}(t)=\left\langle t^{2}, 7 t-12, t^{2}\right\rangle \quad \mathbf{r}_{2}(t)=\left\langle 4 t-3, t^{2}, 5 t-6\right\rangle$$ for \(t \geqslant 0 .\) Do the particles collide?

Step-by-Step Solution

Answer

Yes, the particles collide at \( t = 3 \).

Step 1: Equate the Position Vectors

First, set the position vectors of both particles equal to each other to find a condition where they could potentially collide. We have:\( \mathbf{r}_{1}(t_1) = \langle t_1^2, 7t_1 - 12, t_1^2 \rangle \) \( \mathbf{r}_{2}(t_2) = \langle 4t_2 - 3, t_2^2, 5t_2 - 6 \rangle \)Set the vectors equal to find equations for each component. We need to solve the system for some \( t_1 \) and \( t_2 \) such that all components match.

Step 2: Solve for t in the First Component

Equate the first components of \( \mathbf{r}_{1}(t) \) and \( \mathbf{r}_{2}(t) \):\[ t_1^2 = 4t_2 - 3 \]

Step 3: Solve for t in the Second Component

Equate the second components of \( \mathbf{r}_{1}(t) \) and \( \mathbf{r}_{2}(t) \):\[ 7t_1 - 12 = t_2^2 \]

Step 4: Solve for t in the Third Component

Equate the third components of \( \mathbf{r}_{1}(t) \) and \( \mathbf{r}_{2}(t) \):\[ t_1^2 = 5t_2 - 6 \]

Step 5: Solve the System of Equations

Use the equations from steps 2, 3, and 4 to find common solutions for \( t_1 \) and \( t_2 \).1. From Step 2 equation: \( t_1^2 = 4t_2 - 3 \)2. From Step 4 equation: \( t_1^2 = 5t_2 - 6 \)Equate them: \( 4t_2 - 3 = 5t_2 - 6 \)Solve for \( t_2 \): \( t_2 = 3 \).Substitute \( t_2 = 3 \) back into any of the equations:1. Substituting into \( t_1^2 = 4t_2 - 3 \):\[ t_1^2 = 4(3) - 3 = 9 \]\( t_1 = 3 \) or \( t_1 = -3 \); since \( t_1 \geq 0 \), we have \( t_1 = 3 \).

Step 6: Verify the Solution

Substitute \( t_1 = 3 \) and \( t_2 = 3 \) into the original position vectors to verify\( \mathbf{r}_1(3) = \langle 9, 9, 9 \rangle \)\( \mathbf{r}_2(3) = \langle 9, 9, 9 \rangle \)Since both vectors are equal at \( t = 3 \), the particles collide.

Key Concepts

Vector Functions

Vector functions are mathematical tools used to describe the paths or trajectories of objects moving through space. A vector function assigns a vector to each point in a domain, often representing time. For example, trajectories for two particles given by vector functions

• \( \mathbf{r}_{1}(t)=\left\langle t^{2}, 7 t-12, t^{2}\right\rangle \)
• \( \mathbf{r}_{2}(t)=\left\langle 4 t-3, t^{2}, 5 t-6\right\rangle \)

Here, each vector function provides the position of a particle in 3D space at any time \(t\). Each component of the vector corresponds to a spatial dimension—x, y, or z. Understanding vector functions helps us investigate if and when particles will meet or collide by setting these vector paths equal at certain time points.

System of Equations

A system of equations consists of multiple equations that are solved together to find common variable solutions. When detecting potential collisions, we set the position vectors equal and solve the resulting system of equations.
For the given functions:

• First component: \( t_1^2 = 4t_2 - 3 \)
• Second component: \( 7t_1 - 12 = t_2^2 \)
• Third component: \( t_1^2 = 5t_2 - 6 \)

To determine if the particles collide, find values for \( t_1 \) and \( t_2 \) that satisfy all three equations simultaneously. Solving these sorts of equations often involves substitution or elimination to find a shared solution. For example, equating equations from the first and third components to find \( t_2 = 3 \) and then substituting to find \( t_1 = 3 \). This approach reveals the time when both particles occupy the same position in space.

Position Vectors

Position vectors give the exact location of an object in a coordinate system at a given time. They combine all spatial dimensions into one entity.
For two moving particles, as in our example, the position vectors are functions of time. Setting position vectors equal helps us determine a common point in time and space.
Consider:

• \( \mathbf{r}_{1}(3) = \langle 9, 9, 9 \rangle \)
• \( \mathbf{r}_{2}(3) = \langle 9, 9, 9 \rangle \)

When these vectors are equal, both particles are at the precise same location in space and time. Because particles in our example have identical position vectors at \(t = 3\), they collide. Position vectors are essential in identifying such overlaps and confirming their accuracy in predictions of motion and collision.