At an air show, a jet has a trajectory following the curve y = 3x². When the jet is at the point (5, 75), it has a speed of 550 km/h. Determine the velocity vector at this point. (Express numbers in exact form. Use symbolic notation and fractions where needed. Give your answer using component form or standard basis vectors.) velocity:
At an air show, a jet has a trajectory following the curve y = 3x². When the jet is at the point (5, 75), it has a speed of 550 km/h. Determine the velocity vector at this point. (Express numbers in exact form. Use symbolic notation and fractions where needed. Give your answer using component form or standard basis vectors.) velocity:
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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![### Problem Statement:
At an air show, a jet has a trajectory following the curve \( y = 3x^2 \). When the jet is at the point \( (5, 75) \), it has a speed of 550 km/h. Determine the velocity vector at this point.
(Express numbers in exact form. Use symbolic notation and fractions where needed. Give your answer using component form or standard basis vectors.)
#### Solution:
Determine the velocity:
\[ \text{velocity:} \]
### Explanation:
To solve this problem, we need to find the velocity vector at the specified point and given speed. The velocity vector in a plane can be determined using the parametric form of the position function and differentiating it.
### Finding the Velocity Vector:
1. **Trajectory Equation:**
\[ y = 3x^2 \]
2. **Coordinates at the Given Point:**
\[ (5, 75) \]
3. **Speed:**
\[ \text{Speed} = 550 \text{ km/h} \]
4. **Derivative (velocity components):**
\[ \text{Let } x(t) \text{ and } y(t) \text{ be the parametric equations. Then, } y = 3x^2 \]
\[ \text{Differentiate } y \text{ with respect to } x: \]
\[ \frac{dy}{dx} = 6x \]
5. **Evaluating at \( x = 5 \):**
\[ \frac{dy}{dx} \bigg|_{x=5} = 6 \cdot 5 = 30 \]
6. **Components of the Velocity (normalized):**
\[ \text{Let } V_x \text{ and } V_y \text{ be the x and y components of the velocity vector.} \]
\[ \frac{V_y}{V_x} = 30 \Rightarrow V_y = 30V_x \]
7. **Finding the magnitudes:**
\[ \sqrt{V_x^2 + V_y^2} = 550 \]
\[ \sqrt{V_x^2 + (30V_x)^2} = 550 \]
\[ \sqrt{1 + 900}V_x = 550 \]
\[ \sqrt](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe815a6c0-dcfd-4fc5-aa4e-03e2f1f5f601%2F56447858-055e-4124-94d6-6d37de44810d%2F646qz7q_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
At an air show, a jet has a trajectory following the curve \( y = 3x^2 \). When the jet is at the point \( (5, 75) \), it has a speed of 550 km/h. Determine the velocity vector at this point.
(Express numbers in exact form. Use symbolic notation and fractions where needed. Give your answer using component form or standard basis vectors.)
#### Solution:
Determine the velocity:
\[ \text{velocity:} \]
### Explanation:
To solve this problem, we need to find the velocity vector at the specified point and given speed. The velocity vector in a plane can be determined using the parametric form of the position function and differentiating it.
### Finding the Velocity Vector:
1. **Trajectory Equation:**
\[ y = 3x^2 \]
2. **Coordinates at the Given Point:**
\[ (5, 75) \]
3. **Speed:**
\[ \text{Speed} = 550 \text{ km/h} \]
4. **Derivative (velocity components):**
\[ \text{Let } x(t) \text{ and } y(t) \text{ be the parametric equations. Then, } y = 3x^2 \]
\[ \text{Differentiate } y \text{ with respect to } x: \]
\[ \frac{dy}{dx} = 6x \]
5. **Evaluating at \( x = 5 \):**
\[ \frac{dy}{dx} \bigg|_{x=5} = 6 \cdot 5 = 30 \]
6. **Components of the Velocity (normalized):**
\[ \text{Let } V_x \text{ and } V_y \text{ be the x and y components of the velocity vector.} \]
\[ \frac{V_y}{V_x} = 30 \Rightarrow V_y = 30V_x \]
7. **Finding the magnitudes:**
\[ \sqrt{V_x^2 + V_y^2} = 550 \]
\[ \sqrt{V_x^2 + (30V_x)^2} = 550 \]
\[ \sqrt{1 + 900}V_x = 550 \]
\[ \sqrt
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