4. Solve for the resultant velocity of the vectors below:

A = 3.7m / s @0 degrees

B = 1.5m / s @38 degrees C = 6.1m / s @ 90 degrees

D = 2.2m / s @ 160 degrees

To solve for the resultant velocity of the given vectors, we need to break each vector into its horizontal (\( x \)-component) and vertical (\( y \)-component). Then, we can sum all the components to get the resultant vector in both directions, and finally calculate the magnitude and direction of the resultant vector.

### Step 1: Break each vector into components

- **Vector A**: 
  - Magnitude: 3.7 m/s, Angle: 0°
  - \( A_x = 3.7 \cos(0°) = 3.7 \)
  - \( A_y = 3.7 \sin(0°) = 0 \)

- **Vector B**:
  - Magnitude: 1.5 m/s, Angle: 38°
  - \( B_x = 1.5 \cos(38°) \)
  - \( B_y = 1.5 \sin(38°) \)

- **Vector C**:
  - Magnitude: 6.1 m/s, Angle: 90°
  - \( C_x = 6.1 \cos(90°) = 0 \)
  - \( C_y = 6.1 \sin(90°) = 6.1 \)

- **Vector D**:
  - Magnitude: 2.2 m/s, Angle: 160°
  - \( D_x = 2.2 \cos(160°) \)
  - \( D_y = 2.2 \sin(160°) \)

Now, I'll calculate the exact values for \( B_x \), \( B_y \), \( D_x \), and \( D_y \).

### Step 2: Calculate the components

Let me calculate those.The components of each vector are as follows:

- **Vector A**:
  - \( A_x = 3.7 \)
  - \( A_y = 0.0 \)

- **Vector B**:
  - \( B_x = 1.18 \)
  - \( B_y = 0.92 \)

- **Vector C**:
  - \( C_x = 0.0 \)
  - \( C_y = 6.1 \)

- **Vector D**:
  - \( D_x = -2.07 \)
  - \( D_y = 0.75 \)

### Step 3: Sum the components

To get the resultant vector's components, we add the individual components:

\[
R_x = A_x + B_x + C_x + D_x
\]
\[
R_y = A_y + B_y + C_y + D_y
\]

Now, let's compute the resultant components.The resultant components are:

- \( R_x = 2.81 \)
- \( R_y = 7.78 \)

### Step 4: Calculate the magnitude and direction of the resultant vector

- The magnitude \( R \) is given by:
  \[
  R = \sqrt{R_x^2 + R_y^2}
  \]

- The direction \( \theta \) (relative to the positive \( x \)-axis) is:
  \[
  \theta = \tan^{-1} \left( \frac{R_y}{R_x} \right)
  \]

Let's compute the magnitude and direction.The resultant velocity is:

- **Magnitude**: 8.27 m/s
- **Direction**: 70.10° relative to the positive \( x \)-axis.

Would you like more details or have any questions?

### Related questions:
1. How do you break a vector into components?
2. Why is the angle measured from the positive \( x \)-axis?
3. What are some common applications of vector addition in physics?
4. How can you add more than two vectors graphically?
5. What is the difference between velocity and speed?

### Tip:
Always draw a diagram when working with vectors to visualize the direction and components clearly.

Solve for the resultant velocity of the vectors below:

A = 3.7m/s @ 0 degrees

B = 1.5m/s @ 38 degrees

C = 6.1m/s @ 90 degrees

D = 2.2m/s @ 160 degrees

Find the resultant velocity by adding vectors A = 3.7m/s @ 0 degrees, B = 1.5m/s @ 38 degrees, C = 6.1m/s @ 90 degrees, and D = 2.2m/s @ 160 degrees. Learn how to use vector components, trigonometric ratios, and the Pythagorean Theorem in this step-by-step solution. Suitable for Grades 10-12 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