If two objects travel through space along two different curves, it is often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) Their paths 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 following vector functions. r₁(t) = (t², 13t 42, t²), r₂(t) = (11t - 30, t², 10t - 24) for t 20 Find the values of t at which the particles collide. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
If two objects travel through space along two different curves, it is often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) Their paths 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 following vector functions. r₁(t) = (t², 13t 42, t²), r₂(t) = (11t - 30, t², 10t - 24) for t 20 Find the values of t at which the particles collide. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.4: The Singular Value Decomposition
Problem 49EQ
Related questions
Question
100%
![---
**Understanding Collisions of Objects in Space**
In physics and engineering, determining whether two objects traveling along different curved paths will collide is a significant problem. This is important in various scenarios such as a missile targeting a moving object or avoiding mid-air aircraft collisions. While their paths might intersect, a collision depends on whether they occupy the same position **at the same time**.
Consider the trajectories of two particles defined by the following vector functions:
\[ \mathbf{r_1}(t) = \{t^2, 13t - 42, t^2 \} \]
\[ \mathbf{r_2}(t) = \{11t - 30, t^2, 10t - 24 \} \]
for \( t \geq 0 \).
---
### Task:
**Find the values of \( t \) at which the particles collide.**
(Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
**\( t = \)**
---
By analyzing these functions, we can determine the instances when the particles meet.
1. **Set the components equal to each other:**
\[ t^2 = 11t - 30 \]
\[ 13t - 42 = t^2 \]
\[ t^2 = 10t - 24 \]
2. **Solve the system of equations:**
Proceed by solving these equations for \( t \).
---
This exercise will provide a clear understanding of how to calculate the precise moments in time at which two moving particles will collide in three-dimensional space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fef258a66-0367-4e84-ab3e-34166dbe87ca%2F7bebfef4-c5c5-4ad9-bac7-d76c7b470482%2F0irbvcl_processed.png&w=3840&q=75)
Transcribed Image Text:---
**Understanding Collisions of Objects in Space**
In physics and engineering, determining whether two objects traveling along different curved paths will collide is a significant problem. This is important in various scenarios such as a missile targeting a moving object or avoiding mid-air aircraft collisions. While their paths might intersect, a collision depends on whether they occupy the same position **at the same time**.
Consider the trajectories of two particles defined by the following vector functions:
\[ \mathbf{r_1}(t) = \{t^2, 13t - 42, t^2 \} \]
\[ \mathbf{r_2}(t) = \{11t - 30, t^2, 10t - 24 \} \]
for \( t \geq 0 \).
---
### Task:
**Find the values of \( t \) at which the particles collide.**
(Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
**\( t = \)**
---
By analyzing these functions, we can determine the instances when the particles meet.
1. **Set the components equal to each other:**
\[ t^2 = 11t - 30 \]
\[ 13t - 42 = t^2 \]
\[ t^2 = 10t - 24 \]
2. **Solve the system of equations:**
Proceed by solving these equations for \( t \).
---
This exercise will provide a clear understanding of how to calculate the precise moments in time at which two moving particles will collide in three-dimensional space.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage