Introduction

In the context of a motorcycle ride that involves traveling 12 kilometers due east and 5 kilometers due south, it is important to distinguish between the distance traveled and the displacement. These two concepts are crucial in physics and everyday life, impacting not only navigation but also understanding the nature of motion and space.

Calculating the Distance Traveled

The distance traveled is a scalar quantity that represents the total path length covered by the rider. To find this, simply add the distances covered in each direction:

Distance Calculation

Total distance covered Distance traveled in the east direction Distance traveled in the south direction

Total distance covered 12 km 5 km 17 km

Understanding Displacement

Displacement, on the other hand, is a vector quantity that refers to the straight-line distance from the starting point to the ending point.

Using the Pythagorean Theorem

For a rider who travels 12 kilometers due east and 5 kilometers due south, you can use the Pythagorean theorem to calculate the straight-line displacement. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the 12 kilometers and 5 kilometers form the two legs of the right triangle, and the displacement is the hypotenuse.

Calculating the Straight-Line Distance

[d sqrt{12^2 5^2} sqrt{144 25} sqrt{169} 13 text{ km}]

Interpreting the Results

The total distance traveled by the rider is 17 kilometers, which is the sum of the individual distances covered in each direction. The straight-line distance from the starting point to the endpoint, known as the displacement, is 13 kilometers.

Vector Representation

From a vector perspective, the displacement can also be represented as a single vector that points from the starting point to the final point. The direction of this vector can be described as having a tangent value of 5/12, indicating that the angle of displacement is such that the south-east component is 5 kilometers and the east component is 12 kilometers.

Conclusion

To summarize, when a rider covers 12 kilometers due east and 5 kilometers due south, they travel a total distance of 17 kilometers, but their displacement, or the straight-line distance to the endpoint, is 13 kilometers.

Understanding the differences between distance and displacement is important for applications ranging from GPS navigation to physics problems. By calculating both these quantities, you get a comprehensive understanding of the motion involved in the ride.