A student observed an ant crawling on a wall. He determined the ant's displacement by using a protractor and ruler. The displacements are as follows 2.5 cm east, 3 cm, 35° north of east and 4.1 cm, 18º west of north

Let's analyze the ant's displacement step by step to determine the resultant displacement.

### Given Displacements:
1. **2.5 cm east** 
2. **3 cm, 35° north of east**
3. **4.1 cm, 18° west of north**

### Step 1: Break Down Each Displacement into Components

1. **First displacement: 2.5 cm east**
   - Since it's directly east, there is only an x-component (east) and no y-component (north):
   \[
   x_1 = 2.5 \, \text{cm}, \quad y_1 = 0 \, \text{cm}
   \]

2. **Second displacement: 3 cm, 35° north of east**
   - This displacement can be broken into:
   \[
   x_2 = 3 \times \cos(35^\circ) 
   \]
   \[
   y_2 = 3 \times \sin(35^\circ)
   \]

3. **Third displacement: 4.1 cm, 18° west of north**
   - This displacement can be broken into:
   \[
   x_3 = -4.1 \times \sin(18^\circ) 
   \]
   \[
   y_3 = 4.1 \times \cos(18^\circ)
   \]
   - Note: The x-component is negative because it's west of north.

### Step 2: Calculate the Components

Let’s compute these values:

1. **Second displacement components:**
   \[
   x_2 = 3 \times \cos(35^\circ) \approx 3 \times 0.8192 = 2.4576 \, \text{cm}
   \]
   \[
   y_2 = 3 \times \sin(35^\circ) \approx 3 \times 0.5736 = 1.7208 \, \text{cm}
   \]

2. **Third displacement components:**
   \[
   x_3 = -4.1 \times \sin(18^\circ) \approx -4.1 \times 0.3090 = -1.2669 \, \text{cm}
   \]
   \[
   y_3 = 4.1 \times \cos(18^\circ) \approx 4.1 \times 0.9511 = 3.8995 \, \text{cm}
   \]

### Step 3: Sum the Components

Sum the x and y components to find the total displacement in each direction:

1. **Total x-component:**
   \[
   x_{\text{total}} = x_1 + x_2 + x_3 = 2.5 + 2.4576 - 1.2669 = 3.6907 \, \text{cm}
   \]

2. **Total y-component:**
   \[
   y_{\text{total}} = y_1 + y_2 + y_3 = 0 + 1.7208 + 3.8995 = 5.6203 \, \text{cm}
   \]

### Step 4: Calculate the Magnitude and Direction of the Resultant Displacement

1. **Magnitude of the resultant displacement (R):**
   \[
   R = \sqrt{(x_{\text{total}})^2 + (y_{\text{total}})^2} = \sqrt{(3.6907)^2 + (5.6203)^2} \approx \sqrt{13.6264 + 31.5861} = \sqrt{45.2125} \approx 6.724 \, \text{cm}
   \]

2. **Direction of the resultant displacement (\(\theta\)):**
   \[
   \theta = \tan^{-1} \left(\frac{y_{\text{total}}}{x_{\text{total}}}\right) = \tan^{-1} \left(\frac{5.6203}{3.6907}\right) \approx \tan^{-1} (1.5223) \approx 56.41^\circ \text{ north of east}
   \]

### Final Result:
The ant's resultant displacement is approximately **6.72 cm** at an angle of **56.41° north of east**.

Would you like more details or have any questions?

Here are 8 related questions for further exploration:
1. How would the resultant displacement change if one of the displacements was doubled?
2. How can vector addition be used in real-world navigation, like in sailing or aviation?
3. What is the difference between displacement and distance?
4. How would the calculation change if the directions were given in a different coordinate system?
5. What happens if the angles were given in radians instead of degrees?
6. How does vector subtraction work in a scenario like this?
7. Can you solve for the resultant displacement if one displacement is directly south?
8. What are the implications of vector addition in physics, particularly in force and velocity calculations?

**Tip:** Always verify that the angles are referenced to the correct axis (e.g., north of east, west of north) when resolving vectors into components. This helps in avoiding mistakes in determining the direction of the components.

Learn how to calculate the resultant displacement of an ant crawling on a wall using vector addition. This problem involves breaking down displacements into components, summing vectors, and determining magnitude and direction. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation