A small rocket is fired from a launch pad 15 m above the ground with an initial velocity left angle 300 comma 500 comma 550 right angle ​m/s. A cross wind blowing to the north produces an acceleration of the rocket of 3.5 m divided by s squared. Assume the​ x-axis points​ east, the​ y-axis points​ north, the positive​ z-axis is vertical​ (opposite g), and the ground is horizontal. Answer parts a through d.
Question content area bottom
Part 1
a. Find the velocity and position vectors for tgreater than or equals0.
The velocity vector is v​(t)equalsleft angle nothing comma nothing comma nothing right angle
  
  
  
.

Let's break down the motion of the rocket and find the velocity and position vectors at any time \( t \geq 0 \).

### Given Information:
- Initial velocity vector: \( \mathbf{v_0} = \langle 300, 500, 550 \rangle \) m/s.
- Cross wind acceleration (only affects the north direction, y-axis): \( a_y = 3.5 \, \text{m/s}^2 \).
- Gravitational acceleration \( g = 9.8 \, \text{m/s}^2 \) (acts along the negative z-axis).

We are dealing with a three-dimensional motion along the x, y, and z directions.

### Velocity Vector
The velocity vector \( \mathbf{v}(t) \) at any time \( t \) is found by integrating the acceleration components with respect to time.

1. **In the x-direction (east)**: 
   - There is no acceleration in the x-direction, so the velocity in the x-direction remains constant.
   - \( v_x(t) = 300 \, \text{m/s} \).

2. **In the y-direction (north)**: 
   - The wind produces a constant acceleration \( a_y = 3.5 \, \text{m/s}^2 \), so the velocity in the y-direction changes linearly with time.
   - \( v_y(t) = 500 + 3.5t \, \text{m/s} \).

3. **In the z-direction (vertical)**:
   - The only acceleration is due to gravity, which is constant at \( -g = -9.8 \, \text{m/s}^2 \).
   - \( v_z(t) = 550 - 9.8t \, \text{m/s} \).

Thus, the velocity vector at time \( t \) is:
\[
\mathbf{v}(t) = \langle 300, 500 + 3.5t, 550 - 9.8t \rangle \, \text{m/s}.
\]

### Position Vector
To find the position vector \( \mathbf{r}(t) \), we integrate the velocity vector components with respect to time.

1. **In the x-direction**:
   - The position in the x-direction changes linearly with time because the velocity is constant.
   - \( r_x(t) = 300t \, \text{m} \).

2. **In the y-direction**:
   - The position in the y-direction involves integrating \( v_y(t) = 500 + 3.5t \).
   - \( r_y(t) = 500t + \frac{1}{2}(3.5)t^2 = 500t + 1.75t^2 \, \text{m} \).

3. **In the z-direction**:
   - The position in the z-direction involves integrating \( v_z(t) = 550 - 9.8t \).
   - \( r_z(t) = 550t - \frac{1}{2}(9.8)t^2 + 15 \, \text{m} \).
   - (We add 15 m because the rocket is launched from a height of 15 m above the ground).

Thus, the position vector at time \( t \) is:
\[
\mathbf{r}(t) = \langle 300t, 500t + 1.75t^2, 550t - 4.9t^2 + 15 \rangle \, \text{m}.
\]

### Summary
- **Velocity vector**: \( \mathbf{v}(t) = \langle 300, 500 + 3.5t, 550 - 9.8t \rangle \, \text{m/s} \).
- **Position vector**: \( \mathbf{r}(t) = \langle 300t, 500t + 1.75t^2, 550t - 4.9t^2 + 15 \rangle \, \text{m} \).

Would you like more details or have any questions?

---

Here are 5 related questions for further exploration:
1. How long will the rocket stay in the air before hitting the ground?
2. What is the maximum height reached by the rocket?
3. What is the distance traveled by the rocket in the horizontal plane when it hits the ground?
4. How does the crosswind affect the total horizontal displacement of the rocket?
5. What would be the trajectory equation of the rocket in 3D space?

**Tip:** In 3D motion problems, breaking the motion into components along each axis makes the problem easier to solve systematically.

A small rocket is fired from a launch pad 15 m above the ground with an initial velocity of <300, 500, 550> m/s. A cross wind blowing to the north produces an acceleration of the rocket of 3.5 m/s². Assume the x-axis points east, the y-axis points north, the positive z-axis is vertical (opposite g), and the ground is horizontal. Find the velocity and position vectors for t ≥ 0.

Explore how to calculate the velocity and position vectors of a rocket fired from a launch pad with a cross wind acceleration. This problem involves key concepts in vector calculus, kinematics, and 3D motion, making it suitable for advanced high school or college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation