Vector Worksheet 2 Answers 1. Determine the resultant of: - ppt ... - Free Printable

Problem Statement:

We are tasked with determining the resultant vector \( \mathbf{R} \) of two given vectors:
- \( \mathbf{A} = 30.0 \, \text{N} \) at \( 0^\circ \)
- \( \mathbf{B} = 40.0 \, \text{N} \) at \( 90^\circ \)

Solution:

#### Step 1: Understand the Vectors
- Vector \( \mathbf{A} \) has a magnitude of 30.0 N and is directed along the positive x-axis (at \( 0^\circ \)).
- Vector \( \mathbf{B} \) has a magnitude of 40.0 N and is directed along the positive y-axis (at \( 90^\circ \)).

#### Step 2: Resolve the Vectors into Components
Since \( \mathbf{A} \) is along the x-axis and \( \mathbf{B} \) is along the y-axis, their components are:
- For \( \mathbf{A} \):
\[
A_x = 30.0 \, \text{N}, \quad A_y = 0
\]
- For \( \mathbf{B} \):
\[
B_x = 0, \quad B_y = 40.0 \, \text{N}
\]

#### Step 3: Find the Resultant Vector Components
The resultant vector \( \mathbf{R} \) is the vector sum of \( \mathbf{A} \) and \( \mathbf{B} \). Its components are:
\[
R_x = A_x + B_x = 30.0 + 0 = 30.0 \, \text{N}
\]
\[
R_y = A_y + B_y = 0 + 40.0 = 40.0 \, \text{N}
\]

#### Step 4: Calculate the Magnitude of the Resultant Vector
The magnitude \( |\mathbf{R}| \) is found using the Pythagorean theorem:
\[
|\mathbf{R}| = \sqrt{R_x^2 + R_y^2}
\]
Substitute the values of \( R_x \) and \( R_y \):
\[
|\mathbf{R}| = \sqrt{(30.0)^2 + (40.0)^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50.0 \, \text{N}
\]

#### Step 5: Determine the Direction of the Resultant Vector
The direction \( \theta \) of the resultant vector is given by the tangent of the angle it makes with the positive x-axis:
\[
\tan \theta = \frac{R_y}{R_x}
\]
Substitute the values of \( R_y \) and \( R_x \):
\[
\tan \theta = \frac{40.0}{30.0} = \frac{4}{3} \approx 1.33
\]
To find \( \theta \), take the inverse tangent:
\[
\theta = \tan^{-1}(1.33) \approx 53^\circ
\]

#### Step 6: Express the Resultant Vector
The resultant vector \( \mathbf{R} \) has a magnitude of 50.0 N and is directed at an angle of \( 53^\circ \) from the positive x-axis.

Final Answer:

\[
\boxed{R = 50 \, \text{N} \text{ at } 53^\circ}
\]