Sure, let's solve this problem step-by-step.
1. Understanding the problem:
- The person travels from point A to point B at a speed of 4 meters per second (m/s).
- Then, they return from point B to point A at a speed of 6 meters per second (m/s).
- We need to find the average speed and average acceleration for the entire trip.
2. Choose a distance:
- Let's assume the distance between point A and point B is [tex]\( d \)[/tex] meters. For simplicity, we will assume [tex]\( d = 1 \)[/tex] meter. This will simplify our calculations, yet the results will be generalizable.
3. Calculate the time taken for each segment of the trip:
- The time taken to travel from A to B (denoted as [tex]\( t_{AB} \)[/tex]) can be calculated using the formula:
[tex]\[
t_{AB} = \frac{d}{\text{speed to B}} = \frac{1 \text{ meter}}{4 \text{ m/s}} = 0.25 \text{ seconds}
\][/tex]
- The time taken to travel from B to A (denoted as [tex]\( t_{BA} \)[/tex]) can be calculated similarly:
[tex]\[
t_{BA} = \frac{d}{\text{speed to A}} = \frac{1 \text{ meter}}{6 \text{ m/s}} = 0.1667 \text{ seconds}
\][/tex]
4. Calculate the total distance and total time:
- The total distance traveled is the entire round trip from A to B and back to A, which is:
[tex]\[
\text{Total distance} = 2d = 2 \times 1 = 2 \text{ meters}
\][/tex]
- The total time taken for the entire trip is the sum of the times for each segment:
[tex]\[
\text{Total time} = t_{AB} + t_{BA} = 0.25 \text{ seconds} + 0.1667 \text{ seconds} = 0.4167 \text{ seconds}
\][/tex]
5. Calculate the average speed:
- The average speed is defined as the total distance traveled divided by the total time taken:
[tex]\[
\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{2 \text{ meters}}{0.4167 \text{ seconds}} = 4.8 \text{ m/s}
\][/tex]
6. Calculate the average acceleration:
- Average acceleration is defined as the change in velocity divided by the time taken for the change. Since the person starts and ends at the same point (A), the net displacement is zero. This means that their velocity vector hasn't changed in direction or magnitude at the end compared to the start, meaning:
[tex]\[
\text{Average acceleration} = 0
\][/tex]
Therefore, the average speed during this interval is [tex]\( 4.8 \text{ m/s} \)[/tex], and the average acceleration is [tex]\( 0 \)[/tex].
