A bicycle travels 3.2 km due east in 0.10 h, then 5.6 km at 15.0° east of north in 0.11 h, and finally another 3.2 km due east in 0.10 h to reach its destination. The time lost in turning is negligible. What is the average velocity for the entire trip?
i need to find the magnitude in km/h
and direction in degrees as well as north south west east.
To find the average velocity for the entire trip, we need to calculate the total displacement and the total time taken.
First, let's calculate the total displacement. We can break down the displacement into its x and y components.
The first leg of the journey is a displacement of 3.2 km due east. This can be represented as a displacement vector (3.2 km, 0°).
The second leg of the journey is a displacement of 5.6 km at 15° east of north. To find the x and y components of this displacement, we can use trigonometry. The x-component can be calculated as 5.6 km * cos(15°) and the y-component as 5.6 km * sin(15°).
The third leg of the journey is a displacement of 3.2 km due east. This can be represented as a displacement vector (3.2 km, 0°).
Now, let's calculate the x and y components of the total displacement.
x-component = (3.2 km) + (5.6 km * cos(15°)) + (3.2 km) = 3.2 km + (5.6 km * 0.9659) + 3.2 km = 3.2 km + 5.40304 km + 3.2 km = 11.80304 km
y-component = (5.6 km * sin(15°)) = 5.6 km * 0.2588 = 1.44528 km
To find the magnitude (or the total distance) of the displacement, we can use the Pythagorean theorem:
magnitude = sqrt((x-component)^2 + (y-component)^2) = sqrt((11.80304 km)^2 + (1.44528 km)^2) = sqrt(138.9804653 km^2 + 2.08970139 km^2) = sqrt(141.0701667 km^2) = 11.874 km
Next, we need to calculate the total time taken for the entire trip. This is simply the sum of the individual times taken for each leg of the journey:
total time = (0.10 h) + (0.11 h) + (0.10 h) = 0.31 h
Finally, we can calculate the average velocity by dividing the total displacement by the total time taken:
average velocity = magnitude / total time = 11.874 km / 0.31 h = 38.29 km/h
Therefore, the average velocity for the entire trip is 38.29 km/h. To determine the direction, we need to find the angle of the displacement from the positive x-axis.
angle = tan^(-1)(y-component / x-component) = tan^(-1)(1.44528 km / 11.80304 km) = tan^(-1)(0.1225) = 7.00°
Since the displacement is eastward, the direction would be 7.00° east of north.
In summary, the average velocity for the entire trip is 38.29 km/h, and the direction is 7.00° east of north.