an orienteer runs 300m north and then 400m to the southeast (at a 45 degree angle to north). if he runs at a constant speed, how far away is he from the starting position?
To find the distance of the orienteer from the starting position after running 300m north and then 400m to the southeast at a 45-degree angle to north, we can use the concept of vector addition.
Step 1: Break down the 400m southeast vector into its north and east components.
- The angle between the southeast direction and the north direction is 45 degrees.
- The southeast vector of 400m can be divided into two component vectors, one in the north direction and the other in the east direction.
- Since the southeast direction is at a 45-degree angle to the north, both the north and east components will have the same magnitude.
- Therefore, each component vector will have a magnitude of 400m / √2.
Step 2: Determine the net displacement of the orienteer.
- The north component vector of 400m / √2 and the 300m north vector are in the same direction.
- When vectors are in the same direction, we can add their magnitudes to find the net magnitude.
- The net displacement in the north direction is 400m / √2 + 300m = (400√2 + 300) m.
- The east component vector of 400m / √2 is perpendicular to the north component vector and the 300m north vector.
- When vectors are perpendicular, we can use the Pythagorean theorem to find the net magnitude.
- The net displacement in the east direction is equal to the magnitude of the east component vector, which is also 400m / √2.
Step 3: Calculate the distance from the starting position.
- To find the distance from the starting position, we need to combine the north and east displacements.
- The distance can be found using the Pythagorean theorem with the north displacement as one side and the east displacement as the other side.
- Distance from the starting position = √((400√2 + 300)² + (400√2)²) = √((1600 + 1200√2 + 900) + 320000) = √(320000 + 1200√2)
Therefore, the orienteer is approximately √(320000 + 1200√2) meters away from the starting position.