Final answer:
The car initially drives south and accelerates with a component in the southward direction which must cancel out its southern velocity to drive west. By using vector decomposition, we find the southward component of acceleration and use it to determine the time needed for the car's southward velocity component to reach zero, thus changing direction to directly west.
Step-by-step explanation:
To determine how long it takes for a car initially driving south to change direction and drive directly west, we can use vector decomposition of acceleration and initial velocity. The car drives at 26.8 m/s south and accelerates at 3.75 m/s² at a 155.0° angle. This acceleration angle from the south direction indicates acceleration has both a southward and westward component.
To drive directly west, the southern velocity component must be neutralized by the southward acceleration. We begin by finding the westward component of the acceleration using the formula a_west = a × cos(θ), where a is the magnitude of the acceleration and θ is the angle of acceleration relative to the west direction. In this case, θ is 155.0° - 180° = -25.0° (since west is 180° from north). The negative sign indicates that the acceleration's westward component is actually directed to the east.
Thus, the relevant formula to calculate the time needed for the car's southward velocity to become zero is t = Δv / a, where Δv is the change in velocity and a is the acceleration. Here, the southward acceleration component must equal the initial southern velocity. The eastward acceleration component does not affect the time for the car to start moving directly west, but it will continue to increase the car's speed in the east direction. The precise calculation for the time can be determined through proper vector resolution and the use of trigonometry.