Answer:
The magnitude of the average velocity for the entire trip is approximately 27.80 km/h.
Step-by-step explanation:
To find the magnitude of the average velocity for the entire trip, we need to calculate the total displacement and divide it by the total time taken.
Let's break down the given information step by step:
The bicycle travels 8.20 km due east in 0.300 h. This gives us a displacement of 8.20 km in the +x-direction.
The bicycle then travels 13.60 km at 15.0° east of north in 0.470 h. To find the vertical and horizontal components of this displacement, we can use trigonometry. The horizontal component is given by 13.60 km * cos(15°), and the vertical component is given by 13.60 km * sin(15°).
Horizontal component = 13.60 km * cos(15°)
= 13.60 km * 0.96592582628
≈ 13.133 km
Vertical component = 13.60 km * sin(15°)
= 13.60 km * 0.2588190451
≈ 3.512 km
Therefore, the displacement for this leg of the trip is 13.133 km in the +x-direction and 3.512 km in the +y-direction.
Finally, the bicycle travels another 8.20 km due east in 0.300 h. This gives us a displacement of 8.20 km in the +x-direction.
Now, let's calculate the total displacement by adding up the individual displacements:
Total displacement in the x-direction = 8.20 km + 13.133 km + 8.20 km
≈ 29.533 km
Total displacement in the y-direction = 3.512 km
The total time taken for the entire trip is the sum of the individual times: 0.300 h + 0.470 h + 0.300 h = 1.070 h.
To find the average velocity, we divide the total displacement vector by the total time:
Average velocity in the x-direction = Total displacement in the x-direction / Total time
= 29.533 km / 1.070 h
≈ 27.58 km/h
Average velocity in the y-direction = Total displacement in the y-direction / Total time
= 3.512 km / 1.070 h
≈ 3.28 km/h
The magnitude of the average velocity is the square root of the sum of the squares of the average velocities in each direction:
Magnitude of average velocity = sqrt((Average velocity in the x-direction)^2 + (Average velocity in the y-direction)^2)
= sqrt((27.58 km/h)^2 + (3.28 km/h)^2)
≈ sqrt(761.5764 km^2/h^2 + 10.7584 km^2/h^2)
≈ sqrt(772.3348 km^2/h^2)
≈ 27.80 km/h
Therefore, the magnitude of the average velocity for the entire trip is approximately 27.80 km/h.