To find the average velocity of the train during the entire run, we need to calculate the total displacement and divide it by the total time.
Step 1: Convert time to hours
The time spent moving east is 35.0 minutes, which is equal to 35.0/60 = 0.5833 hours.
The time spent moving in the direction 50.0° east of due north is 10.0 minutes, which is equal to 10.0/60 = 0.1667 hours.
The time spent moving west is 49.0 minutes, which is equal to 49.0/60 = 0.8167 hours.
Step 2: Calculate the displacement in the x-direction (east/west)
Since the train moves east for 35.0 minutes and then moves west for 49.0 minutes, the displacement in the x-direction is 0 km as the eastward and westward displacements cancel each other out.
Step 3: Calculate the displacement in the y-direction (north)
Displacement in the y-direction = distance * sin(angle from north)
The distance traveled in the direction 50.0° east of due north can be calculated by:
distance = speed * time
For the 10 minutes spent traveling in the direction 50.0° east of due north:
distance = 61.0 km/hr * 0.1667 hr
distance = 10.18 km
The displacement in the y-direction is then:
displacement = distance * sin(angle from north)
displacement = 10.18 km * sin(50.0°)
displacement ≈ 7.83 km
So, the displacement in the y-direction is approximately 7.83 km.
Step 4: Calculate the total displacement and time
The total displacement is the vector sum of the x and y displacements. Since the x-direction displacement is 0 km, the total displacement is equal to the displacement in the y-direction.
The total time is the sum of the individual times spent in each direction:
Total time = 0.5833 hr + 0.1667 hr + 0.8167 hr
Total time ≈ 1.5667 hr
Step 5: Calculate average velocity
Average velocity = total displacement / total time
Average velocity = 7.83 km / 1.5667 hr
Average velocity ≈ 5.00 km/h
So, the average velocity of the train during this run is approximately 5.00 km/h.
To find the magnitude of the velocity:
The magnitude of the velocity is the absolute value of the average velocity.
Magnitude of velocity = |5.00 km/h|
Magnitude of velocity = 5.00 km/h
So, the magnitude of the velocity is 5.00 km/h.
To find the direction (counterclockwise from east):
The direction can be found using the inverse tangent function, arctan.
Direction (counterclockwise from east) = arctan(displacement in the y-direction / displacement in the x-direction)
In our case, the displacement in the x-direction is 0 km and the displacement in the y-direction is 7.83 km.
Direction (counterclockwise from east) = arctan(7.83 km / 0 km)
Since the displacement in the x-direction is 0 km, this implies the train has not moved east or west, therefore, it does not have a direction counterclockwise from east.
So, the direction (counterclockwise from east) is undefined.