A person walks first at a constant speed of 5.10 m/s along a straight line from point circled A to point circled B and then back along the line from circled B to circled A at a constant speed of 3.25 m/s.

(a) What is her average speed over the entire trip?
d/delta t

total distance = 2d

total time = d/5.10 + d/3.25
average speed = 2/(1/5.10 + 1/3.25) = 3.97 m/s

To find the average speed over the entire trip, we need to calculate the total distance traveled and divide it by the total time taken.

Let's find the total distance traveled first. We know that the person walks from point A to point B and then back to point A.

Distance from A to B = Distance from B to A

Let's assume that the distance from A to B is 'd'. Therefore, the distance from B to A is also 'd'.

Total distance traveled = Distance from A to B + Distance from B to A
= d + d
= 2d

Now, let's find the time taken for each leg of the trip.

Time taken from A to B (t1) = Distance / Speed
= d / 5.10 m/s

Time taken from B to A (t2) = Distance / Speed
= d / 3.25 m/s

The total time taken (t total) for the round trip is the sum of t1 and t2.

t total = t1 + t2

Now, we can find the average speed (v average) using the formula:

v average = Total distance / Total time
= 2d / (t1 + t2)

Please note that we need to know the value of 'd' to provide the exact average speed value.

To find the average speed of the person over the entire trip, we need to determine the total distance traveled and divide it by the total time taken.

Let's start by finding the distance between points A and B. Since the person walks at a constant speed of 5.10 m/s, we can use the formula:

Distance = Speed × Time

Let's assume it takes the person t seconds to travel from A to B. The distance from A to B is given by:

Distance_AB = Speed_AB × Time_AB
Distance_AB = 5.10 m/s × t

Similarly, when the person walks back from B to A at a constant speed of 3.25 m/s, the distance from B to A is given by:

Distance_BA = Speed_BA × Time_BA
Distance_BA = 3.25 m/s × t

Now, to find the total distance traveled, we can add the distances between A and B and B and A:

Total distance = Distance_AB + Distance_BA
Total distance = 5.10 m/s × t + 3.25 m/s × t

Since the person returns to the starting point (A) after reaching B, the total time taken for the entire trip would be the time taken to go from A to B (t) plus the time taken to go from B to A (t). Therefore:

Total time = Time_AB + Time_BA
Total time = t + t

Now, we can find the average speed:

Average speed = Total distance / Total time
Average speed = (5.10 m/s × t + 3.25 m/s × t) / (t + t)

Simplifying this equation gives us:

Average speed = (8.35 m/s × t) / (2t)
Average speed = 8.35 m/s / 2

Therefore, the average speed of the person over the entire trip is 4.175 m/s.