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 magnitude and angle of the average velocity, we need to find the total displacement and total time taken by the train.
Step 1: Find the total displacement
We'll break down the motion of the train into its east-west and north-south components. Then we can add these components to find the total displacement.
a) East-West component:
The train travels east for 41 minutes and west for 60 minutes. Since the train's velocity is constant at 68.0 km/h, we can calculate the distance traveled using the equation:
Distance = Velocity * Time
Distance east = 68.0 km/h * (41/60) h = 46.7 km
Distance west = 68.0 km/h * (60/60) h = 68.0 km
The displacement in the east-west direction is:
Displacement = Distance east - Distance west = 46.7 km - 68.0 km = -21.3 km (west)
b) North-South component:
The train travels in a direction 65.0° east of due north for 22.0 minutes. To find the distance traveled in this direction, we need to calculate the component of the velocity in the north direction.
Velocity north = Velocity * sin(angle)
Velocity north = 68.0 km/h * sin(65.0°)
Distance north = Velocity north * Time
Distance north = (68.0 km/h * sin(65.0°)) * (22/60) h = 26.6 km (north)
The displacement in the north-south direction is:
Displacement = Distance north = 26.6 km (north)
Now, we add the east-west and north-south displacements to find the total displacement:
Total displacement = Displacement east-west + Displacement north-south
Total displacement = -21.3 km (west) + 26.6 km (north)
Step 2: Find the total time
The total time is the sum of the times taken for each segment of the trip:
Total time = 41 minutes + 22 minutes + 60 minutes = 123 minutes = 2.05 hours
Step 3: Find the average velocity
Average velocity = Total displacement / Total time
a) Magnitude of average velocity:
Magnitude = |Average velocity| = √(east-west displacement^2 + north-south displacement^2) / Total time
Magnitude = √((-21.3 km)^2 + (26.6 km)^2) / 2.05 h ≈ 33.1 km/h
b) Angle of average velocity:
Angle = tan^(-1)(north-south displacement / east-west displacement)
Angle = tan^(-1)(26.6 km / (-21.3 km))
Since we're asked for the angle relative to north, with east of north positive and west of north negative, the angle will be negative because the displacement is westward.
Angle ≈ -52.4°
Therefore, the (a) magnitude of the average velocity is 33.1 km/h and (b) the angle of the average velocity relative to north is approximately -52.4°.