Final answer:
To find Δy when the runner is moving directly east, we analyze the initial northward velocity and acceleration with a south-east component. Using kinematics and trigonometry, we determine the displacement in the north-south direction that corresponds to the point when the northward velocity becomes zero.
Step-by-step explanation:
To determine the change in y-position, or Δy, when the runner is moving directly east, we must consider the initial velocity components and the acceleration components. The runner starts moving north at 2.88 m/s and accelerates at 0.350 m/s² at an angle of -52.0°, which implies acceleration in the south-east direction.
Since her acceleration has a southward component, the runner's northward velocity will be decreasing. We are looking for the change in position (Δy) at the instant she starts moving directly east, meaning all her northward velocity component has been canceled out. To find this, we can use kinematic equations, particularly the following one which relates initial velocity, acceleration, and displacement:
v² = u² + 2aΔs, where:
v is the final velocity in the north-south direction (which would be 0 when moving directly east)
u is the initial velocity in the north-south direction (2.88 m/s to the north)
a is the acceleration in the north-south direction
Δs (Δy in this problem) is the displacement in the north-south direction
To find the y-component of acceleration, we use trigonometry:
a_y = a * cos(θ), where θ is the angle of acceleration
This gives us a_y = 0.350 m/s² * cos(-52.0°), though remember that angles measured from the horizontal axis are taken as positive counter-clockwise, so this would be in the negative y-direction. By rearranging the equation for Δy, we can solve for the displacement just as the runner begins to move directly east. This calculated Δy would be negative, indicating a movement to the south relative to her starting position.