A person walks first at a constant speed of 4.90 m/s along a straight line from point A to point B and then back along the line from B to A at a constant speed of 2.70 m/s.
(a) What is her average speed over the entire trip?
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To find the average speed over the entire trip, we need to find the total distance traveled and the total time taken.
Let's assume the distance from A to B is d.
So, the distance from B to A will also be d.
The time taken to travel from A to B can be calculated using the formula: time = distance / speed.
The time taken to travel from B to A can be calculated using the same formula: time = distance / speed.
Using these formulas, the time taken to travel from A to B is t1 = d / 4.90.
And the time taken to travel from B to A is t2 = d / 2.70.
The total time taken for the entire trip is the sum of t1 and t2: t_total = t1 + t2.
The total distance traveled is the sum of the distances from A to B and from B to A: total distance = d + d = 2d.
The average speed is given by the formula: average speed = total distance / total time.
Substituting the values, we get the average speed: average speed = 2d / (d / 4.90 + d / 2.70).
Simplifying this equation, we get: average speed = 2d / ((2.70*d + 4.90*d)/ (4.90*2.70)).
Simplifying further, we get: average speed = (4.90*2.70*2d) / (2.70*d + 4.90*d).
The d term cancels out, and we are left with: average speed = (4.90*2.70) / (2.70 + 4.90).
Evaluating this expression, we find that the average speed over the entire trip is approximately 3.30 m/s.
To find the average speed, we need to first calculate the total distance traveled and then divide it by the total time taken.
Step 1: Calculate the distance traveled from A to B.
Distance = Speed × Time
Given that the person walks at a constant speed of 4.90 m/s, we need to know the time taken. Let's call this time "t".
Distance (AB) = 4.90 m/s × t
Step 2: Calculate the distance traveled from B to A.
Distance (BA) = 2.70 m/s × t
Step 3: Calculate the total distance.
Total Distance = Distance (AB) + Distance (BA)
Total Distance = 4.90 m/s × t + 2.70 m/s × t
Step 4: Calculate the total time.
The person traveled from A to B and then back from B to A, so the total time taken is the sum of the time spent in each direction.
Total Time = Time (AB) + Time (BA)
Total Time = t + t = 2t
Step 5: Calculate the average speed.
Average Speed = Total Distance / Total Time
Average Speed = (4.90 m/s × t + 2.70 m/s × t) / (2t)
Now, if we simplify the equation, we get:
Average Speed = (7.60 m/s × t) / (2t)
Average Speed = 7.60 m/s / 2
So, the average speed over the entire trip is 3.80 m/s.
It is not the average of 4.9 and 2.7, because she spends more time walking at the slower speed.
Let the total distance between A and B be D
t1 = D/4.9 t2 = D/2.7
Average speed = 2D/*t1 + t2)
= 2D/[(D/4.9) + (D/2.7)]
= 2/[(1/4.9) + 1/(2.7)] = 3.38 m/s