A train at a constant 44.0 km/h moves east for 42 min, then in a direction 45.0° east of due north for 17.0 min, and then west for 51.0 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 of the train during its trip, we need to calculate the total displacement and the total time taken.
Step 1: Convert the time durations to hours.
- 42 minutes is equivalent to 42/60 = 0.7 hours
- 17 minutes is equivalent to 17/60 = 0.28 hours
- 51 minutes is equivalent to 51/60 = 0.85 hours
Step 2: Calculate the eastward displacement.
- The train travels at a constant speed of 44.0 km/h for 0.7 hours.
- Thus, the eastward displacement is 44.0 km/h × 0.7 hours = 30.8 km.
Step 3: Calculate the northward displacement.
- The train travels at a speed of 44.0 km/h in a direction 45.0° east of due north for 0.28 hours.
- The northward component of this displacement can be found using trigonometry: displacement = speed × time × cos(angle).
- The northward displacement equals 44.0 km/h × 0.28 hours × cos(45°) = 10.92 km.
Step 4: Calculate the westward displacement.
- The train travels westward at a speed of 44.0 km/h for 0.85 hours.
- Thus, the westward displacement is 44.0 km/h × 0.85 hours = 37.4 km.
Step 5: Calculate the total displacement.
- The total east-west displacement is the sum of individual displacements, taking into account their directions.
- The total east-west displacement = eastward displacement - westward displacement = 30.8 km - 37.4 km = -6.6 km (since westward is considered negative).
- The total north-south displacement is the sum of individual displacements, taking into account their directions.
- The total north-south displacement = northward displacement = 10.92 km.
Step 6: Calculate the total time taken.
- The train spends 0.7 hours + 0.28 hours + 0.85 hours = 1.83 hours traveling in total.
Step 7: Calculate the average velocity.
- The magnitude of the average velocity is given by the formula: magnitude = total displacement / total time taken.
- magnitude = √(east-west displacement² + north-south displacement²) / total time taken.
- magnitude = √((-6.6 km)² + (10.92 km)²) / 1.83 hours ≈ 9.45 km/h (rounded to two decimal places).
Step 8: Calculate the angle relative to north.
- The angle can be calculated using the formula: angle = arctan(north-south displacement / east-west displacement).
- angle = arctan(10.92 km / -6.6 km) ≈ -59.4° (rounded to one decimal place).
- Since east of north is positive and west of north is negative, the angle relative to north is -59.4°.
Therefore, the (a) magnitude of the average velocity is approximately 9.45 km/h, and (b) the angle relative to north is approximately -59.4°.