An ant moves 4cm [N], then 3 cm [NW], then 8cm [W], and then 5cm on a bearing of 70 degrees.
a) Draw a diagram to show the movements of the ant.
b) Determine the total displacement of the ant from its original position. Find the final resultant vector.
To draw the diagram and determine the total displacement of the ant, we can break down the ant's movements into its x-axis (east-west) and y-axis (north-south) components.
a) Drawing the Diagram:
To start, let's establish a coordinate system. Let the positive x-axis (eastward) be towards the right, and the positive y-axis (northward) be towards the top.
1. The ant moves 4 cm north (up) in the y-axis direction (N).
2. Then, the ant moves 3 cm northwest (NW). This movement can be divided into two components: one in the x-axis and one in the y-axis. To determine these components:
- The x-axis component is the cosine of the angle made by the NW direction with the x-axis. Since NW is a 45-degree angle with both the x and y-axes, the x-axis component will be 3 cm * cos(45°) = 3 cm * 0.707 = 2.121 cm (approximately).
- The y-axis component is the sine of the angle made by the NW direction with the y-axis. Again, since NW is a 45-degree angle, the y-axis component will also be 3 cm * sin(45°) = 3 cm * 0.707 = 2.121 cm (approximately).
3. Next, the ant moves 8 cm west (left) in the x-axis direction (W).
4. Finally, the ant moves 5 cm at a bearing of 70 degrees. Again, we break this movement into its x-axis and y-axis components.
- The x-axis component is 5 cm * cos(70°).
- The y-axis component is 5 cm * sin(70°).
b) Determining the Total Displacement:
Now, we can calculate the total displacement of the ant by summing up all the x and y components separately.
Summing up the x-axis (east-west) components:
Total x-axis displacement = (0 cm) + (-2.121 cm) + (-8 cm) + (5 cm * cos(70°))
Summing up the y-axis (north-south) components:
Total y-axis displacement = (4 cm) + (2.121 cm) + (0 cm) + (5 cm * sin(70°))
To find the final resultant vector, we use the Pythagorean theorem on the total x and y displacements:
Resultant displacement = sqrt((Total x-axis displacement)^2 + (Total y-axis displacement)^2)
By plugging in the calculated values, you can determine the total displacement and the final resultant vector.