91Ó°ÊÓ

Displacement is a measure of how far an object is from its starting point and in which direction. It considers the starting and ending positions, not the path taken. In this exercise, displacement is vital in figuring out the average velocity.
Imagine you are driving a car north, then south. The car's movement north adds to the displacement, while moving south takes away from it, because you're moving in the opposite direction.

• Northward Displacement: The trip takes three-fourth of the total time, moving at 27 m/s. The northward displacement is calculated by multiplying speed and time: \( x_{north} = 27 \text{ m/s} \times \frac{3}{4}t = \frac{81}{4}t \text{ m} \).
• Southward Displacement: During the one-fourth time period heading south, at 17 m/s. This displacement is \( x_{south} = 17 \text{ m/s} \times \frac{1}{4}t = \frac{17}{4}t \text{ m} \).

Finally, the total displacement is the difference between these two, calculated as \( \Delta x = \frac{81}{4}t - \frac{17}{4}t = 16t \text{ m} \), showing net movement to the north.

Northward and southward velocities represent how fast an object moves in each specific direction. In this problem, understanding these speeds is crucial to calculating displacement and average velocity.

• Northward Velocity: The car travels north with an average velocity of 27 m/s. It's moving north for three-quarters of the trip duration. This longer period northward means it greatly affects the overall displacement and average velocity calculations.
• Southward Velocity: The car moves south at 17 m/s, but only for one-quarter of the time. This velocity contributes less to the total displacement because it is both slower and for a shorter time frame.

Ultimately, the faster and longer northward movement results in a net positive displacement, indicating that the car ends up further north than its starting point after the trip.

Calculating the travel time is crucial as it allows us to determine both the displacements and ultimately the average velocity. Travel time directly influences the resulting displacements due to how long the car travels in each direction.
Assume the entire trip duration is denoted by \( t \). Here:

• Northward Time: The car is traveling north for three-fourth of total time, or \( \frac{3}{4}t \).
• Southward Time: The car travels south for one-fourth of the time, or \( \frac{1}{4}t \).

This arrangement is important because it sets the foundation for calculating the separate displacements. The time spent going north is key as it accounts for more influence over the final displacement compared to the southward time segment. Once you have the total travel time \( t \), you can substitute it into the formula for average velocity to determine how fast and in which direction the car traveled on average over the entire distance.