A train at a constant 68.0 km/h moves east for 41 min, then in a direction 65.0° east of due north for 22.0 min, and then west for 60.0 min. What are the (a) magnitude (in km/h) and (b) angle (relative to north, with east of north positive and west of north negative) of its average velocity during this trip?
To find the average velocity, we need to calculate the total displacement and the total time.
First, let's find the total displacement:
For the eastward motion, the train travels at a constant speed of 68.0 km/h for 41 min. The distance traveled can be calculated by multiplying the speed by the time:
Distance = Speed × Time
Distance = 68.0 km/h × (41/60) h
Distance = 46.6 km (rounded to one decimal place)
Next, let's find the displacement for the motion in a direction 65.0° east of due north. Since the train is not moving due north, we need to find the north and east components of this motion.
North Component = Speed × Time × sin(angle)
North Component = 68.0 km/h × (22/60) h × sin(65.0°)
North Component ≈ 22.9 km (rounded to one decimal place)
East Component = Speed × Time × cos(angle)
East Component = 68.0 km/h × (22/60) h × cos(65.0°)
East Component ≈ 11.8 km (rounded to one decimal place)
Now, since the train is moving east, the eastward displacement will be positive. However, since the train is moving east of due north, the northward displacement will be positive as well.
Total northward displacement ≈ 46.6 km + 22.9 km = 69.5 km (rounded to one decimal place)
Total eastward displacement ≈ 11.8 km
Lastly, let's find the total time:
Total time = Time eastward + Time in a direction east of due north + Time westward
Total time = 41 min + 22 min + 60 min
Total time = 123 min (rounded to the nearest minute)
Now we have the total displacement (northward and eastward) and the total time.
To find the magnitude (speed) of the average velocity, divide the total displacement by the total time:
Magnitude of average velocity = Total displacement / Total time
Magnitude of average velocity = √((Total northward displacement^2) + (Total eastward displacement^2)) / Total time
Magnitude of average velocity = √((69.5 km^2) + (11.8 km^2)) / 123 min
Calculating this will give you the magnitude (a) of the average velocity.
To find the angle (b), use the inverse tangent function:
Angle = arctan(Total northward displacement / Total eastward displacement)
Calculate this using the values we found earlier to get the angle (b) relative to north.
Remember to convert the time from minutes to hours for consistency if needed.