Solved PHYS-103 Name: Physics: Vector Addition Worksheet | Chegg.com - Free Printable

Problem Analysis:

The worksheet involves vector addition, which requires breaking vectors into their x and y components and then summing them. Let's solve the problem step by step.

---

Part I: Find the x and y components of each of the following vectors

#### Vector 1:
- Magnitude: \( 12 \, \text{m} \)
- Angle: \( 30^\circ \) (measured counterclockwise from the positive x-axis)

The x-component (\( x_1 \)) and y-component (\( y_1 \)) are calculated using:
\[
x = \text{Magnitude} \cdot \cos(\theta)
\]
\[
y = \text{Magnitude} \cdot \sin(\theta)
\]

For Vector 1:
\[
x_1 = 12 \cdot \cos(30^\circ) = 12 \cdot \frac{\sqrt{3}}{2} = 12 \cdot 0.866 = 10.39 \, \text{m}
\]
\[
y_1 = 12 \cdot \sin(30^\circ) = 12 \cdot \frac{1}{2} = 12 \cdot 0.5 = 6 \, \text{m}
\]

#### Vector 2:
- Magnitude: \( 25 \, \text{m/s} \)
- Angle: \( 60^\circ \) (measured counterclockwise from the positive x-axis)

For Vector 2:
\[
x_2 = 25 \cdot \cos(60^\circ) = 25 \cdot \frac{1}{2} = 25 \cdot 0.5 = 12.5 \, \text{m/s}
\]
\[
y_2 = 25 \cdot \sin(60^\circ) = 25 \cdot \frac{\sqrt{3}}{2} = 25 \cdot 0.866 = 21.65 \, \text{m/s}
\]

#### Vector 3:
- Magnitude: \( 15 \, \text{m/s} \)
- Angle: \( 150^\circ \) (measured counterclockwise from the positive x-axis)

For Vector 3:
\[
x_3 = 15 \cdot \cos(150^\circ) = 15 \cdot \cos(180^\circ - 30^\circ) = 15 \cdot (-\cos(30^\circ)) = 15 \cdot (-0.866) = -12.99 \, \text{m/s}
\]
\[
y_3 = 15 \cdot \sin(150^\circ) = 15 \cdot \sin(180^\circ - 30^\circ) = 15 \cdot \sin(30^\circ) = 15 \cdot 0.5 = 7.5 \, \text{m/s}
\]

---

Part II: Add the following vectors

#### Vector Addition 1:
- Vector 1: \( 12 \, \text{m/s} \) at \( 70^\circ \)
- Vector 2: \( 12 \, \text{m/s} \) at \( 70^\circ \)

##### Step 1: Find the components of Vector 1
\[
x_1 = 12 \cdot \cos(70^\circ) = 12 \cdot 0.342 = 4.104 \, \text{m/s}
\]
\[
y_1 = 12 \cdot \sin(70^\circ) = 12 \cdot 0.940 = 11.28 \, \text{m/s}
\]

##### Step 2: Find the components of Vector 2
\[
x_2 = 12 \cdot \cos(70^\circ) = 12 \cdot 0.342 = 4.104 \, \text{m/s}
\]
\[
y_2 = 12 \cdot \sin(70^\circ) = 12 \cdot 0.940 = 11.28 \, \text{m/s}
\]

##### Step 3: Add the components
\[
x_{\text{net}} = x_1 + x_2 = 4.104 + 4.104 = 8.208 \, \text{m/s}
\]
\[
y_{\text{net}} = y_1 + y_2 = 11.28 + 11.28 = 22.56 \, \text{m/s}
\]

#### Vector Addition 2:
- Vector 1: \( 50 \, \text{m} \) at \( 135^\circ \)
- Vector 2: \( 30 \, \text{m} \) at \( 90^\circ \)

##### Step 1: Find the components of Vector 1
\[
x_1 = 50 \cdot \cos(135^\circ) = 50 \cdot \cos(180^\circ - 45^\circ) = 50 \cdot (-\cos(45^\circ)) = 50 \cdot (-0.707) = -35.35 \, \text{m}
\]
\[
y_1 = 50 \cdot \sin(135^\circ) = 50 \cdot \sin(180^\circ - 45^\circ) = 50 \cdot \sin(45^\circ) = 50 \cdot 0.707 = 35.35 \, \text{m}
\]

##### Step 2: Find the components of Vector 2
\[
x_2 = 30 \cdot \cos(90^\circ) = 30 \cdot 0 = 0 \, \text{m}
\]
\[
y_2 = 30 \cdot \sin(90^\circ) = 30 \cdot 1 = 30 \, \text{m}
\]

##### Step 3: Add the components
\[
x_{\text{net}} = x_1 + x_2 = -35.35 + 0 = -35.35 \, \text{m}
\]
\[
y_{\text{net}} = y_1 + y_2 = 35.35 + 30 = 65.35 \, \text{m}
\]

---

Final Answers:

#### Part I:
1. Vector 1:
\[
x = 10.39 \, \text{m}, \quad y = 6 \, \text{m}
\]
2. Vector 2:
\[
x = 12.5 \, \text{m/s}, \quad y = 21.65 \, \text{m/s}
\]
3. Vector 3:
\[
x = -12.99 \, \text{m/s}, \quad y = 7.5 \, \text{m/s}
\]

#### Part II:
1. Vector Addition 1:
\[
x_{\text{net}} = 8.208 \, \text{m/s}, \quad y_{\text{net}} = 22.56 \, \text{m/s}
\]
2. Vector Addition 2:
\[
x_{\text{net}} = -35.35 \, \text{m}, \quad y_{\text{net}} = 65.35 \, \text{m}
\]

---

Boxed Final Answers:

\[
\boxed{
\begin{aligned}
&\text{Part I:} \\
&1. \, x = 10.39 \, \text{m}, \, y = 6 \, \text{m} \\
&2. \, x = 12.5 \, \text{m/s}, \, y = 21.65 \, \text{m/s} \\
&3. \, x = -12.99 \, \text{m/s}, \, y = 7.5 \, \text{m/s} \\
&\text{Part II:} \\
&1. \, x_{\text{net}} = 8.208 \, \text{m/s}, \, y_{\text{net}} = 22.56 \, \text{m/s} \\
&2. \, x_{\text{net}} = -35.35 \, \text{m}, \, y_{\text{net}} = 65.35 \, \text{m}
\end{aligned}
}
\]