To find the car's displacement from its original position, we need to determine its final position relative to its starting point. We'll break down the movements into their north-south and east-west components.
- Movement North: The car travels 25 miles north.
- Movement West: The car then travels 15 miles west.
- Movement South: Finally, the car travels 10 miles south.
Next, we can calculate the net north-south and east-west displacements:
-
North-South Displacement:
- The car moved 25 miles north and then 10 miles south.
- Net north-south displacement = 25 miles north - 10 miles south = 15 miles north.
-
East-West Displacement:
- The car moved 15 miles west.
- There is no eastward movement, so the net east-west displacement = 15 miles west.
Now, we have:
- North-South displacement = 15 miles north.
- East-West displacement = 15 miles west.
Now, we can find the overall displacement using the Pythagorean theorem, since the displacements form a right triangle:
\[ \text{Displacement} = \sqrt{(\text{North-South Displacement})^2 + (\text{East-West Displacement})^2} \]
Substituting the values we found:
\[ \text{Displacement} = \sqrt{(15 \text{ miles})^2 + (15 \text{ miles})^2} \] \[ = \sqrt{225 + 225} = \sqrt{450} \approx 21.21 \text{ miles} \]
To find the direction of the displacement, we can use the tangent function:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{15}{15} = 1 \] \[ \theta = \tan^{-1}(1) = 45^\circ \]
Since the displacement is towards the northwest (15 miles north and 15 miles west), we can say that the angle is 45 degrees from the north toward the west.
Thus, the car's displacement from its original position is approximately 21.21 miles at an angle of 45 degrees northwest.