Understanding the Resultant Displacement of a Motorcycle Journey
Imagine a scenario where a motorcycle travels 5 kilometers west and then turns north, traveling another 8 kilometers. To find the resultant displacement and its direction, we can use several geometric and trigonometric principles. This article will walk you through the process, utilizing the Pythagorean theorem and the tangent function.
Magnitude of the Resultant Displacement
The first step is to calculate the magnitude of the resultant displacement. When a motorcycle travels 5 kilometers west and then 8 kilometers north, it forms a right triangle. The legs of this triangle are 5 km and 8 km, and the resultant displacement is the hypotenuse.
Using the Pythagorean theorem, the formula to find the hypotenuse (R) is:
[ R sqrt{x^2 y^2} ]Where:- ( x 5 ) km (displacement in the west direction on the x-axis)- ( y 8 ) km (displacement in the north direction on the y-axis)Substituting the given values:[ R sqrt{5^2 8^2} sqrt{25 64} sqrt{89} approx 9.43 text{ km} ]
Direction of the Resultant Displacement
Next, we need to determine the direction of the resultant displacement. This is done by calculating the angle with respect to the west direction. The tangent function helps in finding this formula for the angle ((theta)) is:[ tan theta frac{y}{x} ]Substituting the values:[ tan theta frac{8}{5} ]Using the inverse tangent (arctangent) function to find the angle:[ theta tan^{-1} left(frac{8}{5}right) ]This calculation provides:[ theta approx tan^{-1}(1.6) approx 58^circ ]Since the angle (theta) is measured from the west direction, the resultant displacement is approximately 58° north of west.
Geometric Representation and Calculation Verification
To further understand this concept, one can draw the triangle and apply the Pythagorean theorem to find the magnitude of the resultant displacement.1. Draw a right triangle where one leg is 5 km (west) and the other leg is 8 km (north).2. Use the Pythagorean theorem to find the length of the hypotenuse (resultant displacement):[ R sqrt{5^2 8^2} sqrt{25 64} sqrt{89} approx 9.43 text{ km} ]3. Use the tangent function to find the angle:[ tan theta frac{8}{5} ][ theta tan^{-1} left(frac{8}{5}right) approx 58^circ ]
Conclusion
In conclusion, the magnitude of the resultant displacement when a motorcycle travels 5 km west and then 8 km north is approximately 9.43 kilometers. The direction of this displacement is approximately 58° north of west. Understanding these principles not only helps in solving real-world navigation problems but also in various fields such as physics and engineering.
Key Takeaways
- The magnitude of the resultant displacement can be calculated using the Pythagorean theorem.- The direction of the resultant displacement can be determined using the tangent function.- Applying these principles helps in solving complex navigation and positioning problems.Understanding these concepts is crucial for anyone working with vector quantities in physics and engineering, making this a fundamental topic to master.