An ant travels 2.78 cm (West) and then turns and travels 6.25cm (south 40 degrees East). What is the ant's total displacement?
To calculate the ant's total displacement, we will use vector addition.
First, we need to break down the ant's displacement into its horizontal (East-West) and vertical (North-South) components.
For the Westward displacement of 2.78 cm, we know it is entirely in the West direction, so the horizontal component is -2.78 cm (negative because it is in the opposite direction as East).
For the South 40 degrees East displacement of 6.25 cm, we can find the horizontal and vertical components using trigonometry.
The horizontal component is given by:
horizontal component = displacement * cos(angle)
= 6.25 cm * cos(40 degrees)
≈ 4.76 cm
The vertical component is given by:
vertical component = displacement * sin(angle)
= 6.25 cm * sin(40 degrees)
≈ 3.99 cm
Now, we can add the horizontal and vertical components together to find the total displacement.
Total horizontal displacement = -2.78 cm + 4.76 cm = 1.98 cm (West)
Total vertical displacement = 3.99 cm (South)
Using the Pythagorean theorem, we can find the magnitude (total distance) of the displacement:
Magnitude = sqrt((total horizontal displacement)^2 + (total vertical displacement)^2)
= sqrt((1.98 cm)^2 + (3.99 cm)^2)
≈ 4.53 cm
Therefore, the ant's total displacement is approximately 4.53 cm, in a direction towards the South and slightly to the West.
To find the ant's total displacement, we can use vector addition. First, let's break down the ant's motion into components.
The ant originally travels 2.78 cm west, which we can assign a negative x-component since it's in the opposite direction of the positive x-axis.
The ant then travels 6.25 cm south 40 degrees east. We need to find the east and south components for this motion. To do that, we can use trigonometric functions.
The east component = 6.25 cm * cos(40°)
The south component = -6.25 cm * sin(40°)
Now we add up the component vectors:
Total x-component = -2.78 cm + (6.25 cm * cos(40°))
Total y-component = (-6.25 cm * sin(40°))
Finally, we can find the magnitude (or total displacement) using the Pythagorean theorem:
Total displacement = sqrt((Total x-component)^2 + (Total y-component)^2)
By substituting the values, we can calculate the total displacement of the ant.