https://file.aiquickdraw.com/mathbot/file/1726027177363ulfqjzcy.jpg

Let's solve this problem step by step.

We are tasked with finding the displacement of the roller coaster after it moves in three different directions. We are given:

1. A horizontal movement of \( 2.00 \times 10^2 \) ft.
2. An upward movement of 135 ft at an angle of 30.0° above the horizontal.
3. A downward movement of 135 ft at an angle of 40.0° below the horizontal.

We can approach this problem by breaking each segment into horizontal and vertical components, and then use vector addition to find the total displacement.

### Step 1: Horizontal and Vertical Components

#### First segment:
- This is a purely horizontal movement of \( 2.00 \times 10^2 \) ft, so:
  \[
  \text{Horizontal component} = 200 \, \text{ft}
  \]
  \[
  \text{Vertical component} = 0 \, \text{ft}
  \]

#### Second segment:
- This is 135 ft at an angle of 30.0° above the horizontal, so we use trigonometry to find the components:
  \[
  \text{Horizontal component} = 135 \cdot \cos(30.0^\circ)
  \]
  \[
  \text{Vertical component} = 135 \cdot \sin(30.0^\circ)
  \]

#### Third segment:
- This is 135 ft at an angle of 40.0° below the horizontal, so we again use trigonometry:
  \[
  \text{Horizontal component} = 135 \cdot \cos(40.0^\circ)
  \]
  \[
  \text{Vertical component} = -135 \cdot \sin(40.0^\circ)
  \]
  (The vertical component is negative because it's moving downward.)

### Step 2: Sum the Components

We sum all the horizontal and vertical components from the three segments to find the total horizontal and vertical displacement.

### Step 3: Displacement Magnitude

Finally, the magnitude of the displacement is found using the Pythagorean theorem:
\[
\text{Displacement} = \sqrt{(\text{Total Horizontal Component})^2 + (\text{Total Vertical Component})^2}
\]

Let's compute the exact values. I'll calculate these components now.The total displacement of the roller coaster from its starting point is approximately:

- **Horizontal displacement**: 420.33 ft
- **Vertical displacement**: -19.28 ft (slightly downward)
- **Magnitude of displacement**: 420.77 ft

This gives the overall straight-line distance from the start to the end of the movement.

Would you like further clarification or details on any part of this solution?

Here are five related questions for further exploration:
1. How would the displacement change if the angles of ascent and descent were steeper?
2. What happens to the vertical displacement if both angles were above the horizontal?
3. How could you visualize this displacement on a graph?
4. How would the displacement change if the second or third segments had different distances?
5. How would the answer change if the roller coaster also had a curved segment?

**Tip**: When solving vector problems, always break down movements into horizontal and vertical components and use the Pythagorean theorem to find the magnitude.

A roller coaster moves 200 ft horizontally and then rises 135 ft at an angle of 30.0° above the horizontal. Next, it travels 135 ft at an angle of 40.0° below the horizontal. Use graphical techniques to find the roller coaster's displacement from its starting point to the end of this movement.

Learn how to find the displacement of a roller coaster that moves horizontally, ascends at an angle, and descends at a different angle. Using vector components, trigonometry, and the Pythagorean theorem, this problem explores horizontal and vertical motion, making it suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation