Question

A runner is training for an upcoming marathon by running around a 100-m-diameter circular track at constant speed. Let a coordinate system have its origin at the center of the circle with the x-axis pointing east and the y-axis north. The runner starts at (x,y) = (50m, 0m) and runs 2.5 times around the track in aclockwise direction. What is his displacement vector? Give your answer as a magnitude and direction.

Accepted Answer

Step 1: Understand the problem. The runner is moving in a circular path with a diameter of 100 m, which means the radius of the circle is 50 m. The runner starts at the point (50 m, 0 m) on the positive x-axis and runs 2.5 times around the circle in a clockwise direction. We need to find the displacement vector, which is the straight-line distance and direction from the starting point to the final point. Step 2: Calculate the total angular displacement. One full circle corresponds to an angular displacement of 2π radians. Since the runner completes 2.5 revolutions, the total angular displacement is 2.5 × 2π radians. This equals 5π radians. Since the motion is clockwise, the angular displacement is negative: -5π radians. Step 3: Determine the final angular position. The runner starts at an angle of 0 radians (on the positive x-axis). To find the final angular position, add the total angular displacement to the initial angle. Since -5π radians is equivalent to -π radians (after subtracting multiples of 2π to bring it within one full circle), the final angular position is -π radians. This corresponds to the point (-50 m, 0 m) on the negative x-axis. Step 4: Calculate the displacement vector. The displacement vector is the straight-line vector from the starting point (50 m, 0 m) to the final point (-50 m, 0 m). The x-component of the displacement is Δx = -50 m - 50 m = -100 m, and the y-component is Δy = 0 m - 0 m = 0 m. Thus, the displacement vector is (-100 m, 0 m). Step 5: Find the magnitude and direction of the displacement vector. The magnitude of the displacement vector is given by the formula: Δx2 + Δy2. Substituting Δx = -100 m and Δy = 0 m, the magnitude is 100 m. The direction is the angle of the vector relative to the positive x-axis, which is 180° (or π radians) since the vector points directly to the left.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Displacement

Circular Motion

Vector Components