Find the velocity vector v(t), given the acceleration vector a(t) = (3e¹, (Use symbolic notation and fractions where needed. Give your answer in the vector form.) v(r) = 2(e-2)i + (3r-2)j + (3² +9r+2)k 9) and the initial velocit Incorrect
Find the velocity vector v(t), given the acceleration vector a(t) = (3e¹, (Use symbolic notation and fractions where needed. Give your answer in the vector form.) v(r) = 2(e-2)i + (3r-2)j + (3² +9r+2)k 9) and the initial velocit Incorrect
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![## Finding the Velocity Vector \( \mathbf{v}(t) \)
### Problem Statement:
Given the acceleration vector \( \mathbf{a}(t) = \left(3e^t, 7, 10t + 9\right) \) and the initial velocity \( \mathbf{v}(0) = \left(8, -5, 3\right) \), find the velocity vector \( \mathbf{v}(t) \).
### Solution Strategy:
To find the velocity vector \( \mathbf{v}(t) \), we need to integrate the acceleration vector \( \mathbf{a}(t) \).
1. **Integrate each component of \( \mathbf{a}(t) \):**
- The i-component: \( 3e^t \)
- The j-component: \( 7 \)
- The k-component: \( 10t + 9 \)
2. **Apply initial conditions to determine the constants of integration.**
### Calculation:
1. **Integrate the i-component:**
\[
\int 3e^t \, dt = 3e^t + C_1
\]
2. **Integrate the j-component:**
\[
\int 7 \, dt = 7t + C_2
\]
3. **Integrate the k-component:**
\[
\int (10t + 9) \, dt = 5t^2 + 9t + C_3
\]
4. **Form the general solution for \( \mathbf{v}(t) \):**
\[
\mathbf{v}(t) = (3e^t + C_1)\mathbf{i} + (7t + C_2)\mathbf{j} + (5t^2 + 9t + C_3)\mathbf{k}
\]
5. **Apply the initial condition \( \mathbf{v}(0) = (8, -5, 3) \):**
- i-component: \( 3e^0 + C_1 = 8 \)
\[
3 \cdot 1 + C_1 = 8 \implies C_1 = 5
\]
- j-component: \( 7 \cdot 0 + C_2 = -5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F460d6b43-992c-4555-b09d-3f6791ba763b%2F49efe3a2-bc29-45c9-a6c3-ddf8a63afa88%2Fom5wlw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Finding the Velocity Vector \( \mathbf{v}(t) \)
### Problem Statement:
Given the acceleration vector \( \mathbf{a}(t) = \left(3e^t, 7, 10t + 9\right) \) and the initial velocity \( \mathbf{v}(0) = \left(8, -5, 3\right) \), find the velocity vector \( \mathbf{v}(t) \).
### Solution Strategy:
To find the velocity vector \( \mathbf{v}(t) \), we need to integrate the acceleration vector \( \mathbf{a}(t) \).
1. **Integrate each component of \( \mathbf{a}(t) \):**
- The i-component: \( 3e^t \)
- The j-component: \( 7 \)
- The k-component: \( 10t + 9 \)
2. **Apply initial conditions to determine the constants of integration.**
### Calculation:
1. **Integrate the i-component:**
\[
\int 3e^t \, dt = 3e^t + C_1
\]
2. **Integrate the j-component:**
\[
\int 7 \, dt = 7t + C_2
\]
3. **Integrate the k-component:**
\[
\int (10t + 9) \, dt = 5t^2 + 9t + C_3
\]
4. **Form the general solution for \( \mathbf{v}(t) \):**
\[
\mathbf{v}(t) = (3e^t + C_1)\mathbf{i} + (7t + C_2)\mathbf{j} + (5t^2 + 9t + C_3)\mathbf{k}
\]
5. **Apply the initial condition \( \mathbf{v}(0) = (8, -5, 3) \):**
- i-component: \( 3e^0 + C_1 = 8 \)
\[
3 \cdot 1 + C_1 = 8 \implies C_1 = 5
\]
- j-component: \( 7 \cdot 0 + C_2 = -5
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