Final answer:
The runner's velocity in the x-direction is approximately 2.76 m/s, and in the y-direction it is about 1.25 m/s. Their total velocity is the given speed of 3.05 m/s at a 24.0° angle counterclockwise from the positive x-axis. Assuming constant speed and direction, the runner's acceleration is 0 m/s².
Step-by-step explanation:
The cross-country runner's problem is based on decomposing the velocity vector into its horizontal and vertical components using trigonometrical functions. Let's solve each part of the question:
a. What is the runner's velocity in the x-direction?
To find the runner's velocity in the x-direction, we apply cosine because it represents the adjacent side in a right triangle formed by the velocity vector. Using the formula v_x = v × cos(θ), where v_x is the velocity in the x-direction, v is the speed (3.05 m/s), and θ is the angle (24.0°), we get v_x = 3.05 m/s × cos(24.0°)≈ 2.76 m/s.
b. What is the runner's velocity in the y-direction?To calculate the y-component, we apply sine, which represents the opposite side. The formula is v_y = v × sin(θ), yielding v_y = 3.05 m/s × sin(24.0°)≈ 1.25 m/s.
c. What is the runner's total velocity?
The runner's total velocity is the original speed vector with magnitude 3.05 m/s, pointing 24.0° counterclockwise from the positive x-axis as given.
d. What is the runner's acceleration?
If the runner maintains the given speed and direction, the acceleration is 0 m/s², as there is no change in the magnitude or direction of the velocity.