A train moving at a constant speed of 68.0 km/h moves east for 36.0 min, then in a direction 48.0° east of due north for 30.0 min, and then west for 45.0 min. What is the average velocity of the train during this run?
Find its end position.
End position=68*36/60 E + 68*1/2*(Cos48 N + sin48E)- 68*45/60 E
Average veocity= endposition-start position/ timetotal and the start position is zero.
To find the average velocity of the train during this run, we need to calculate the total displacement and divide it by the total time taken.
First, we need to find the displacement of the train in the east direction. The train moves east for 36.0 minutes at a speed of 68.0 km/h (converted to km/min):
Displacement in east = (Speed east) × (Time east) = 68.0 km/h × (36.0 min / 60 min) = 40.8 km
Next, we need to find the displacement of the train in the north direction. The train moves in a direction 48.0° east of due north for 30.0 minutes. To calculate the displacement in the north direction, we first need to find the component of the velocity in the north direction:
Velocity north = (Speed north) × cos(48.0°) = (68.0 km/h) × cos(48.0°) = 46.5 km/h
Now we can calculate the displacement in the north direction:
Displacement in north = (Velocity north) × (Time north) = 46.5 km/h × (30.0 min / 60 min) = 23.25 km
Lastly, we need to find the displacement in the west direction. The train moves west for 45.0 minutes at a speed of 68.0 km/h:
Displacement in west = (Speed west) × (Time west) = 68.0 km/h × (45.0 min / 60 min) = 51.0 km
Now we can find the total displacement by adding the displacement values in each direction:
Total displacement = Displacement in east + Displacement in north - Displacement in west = 40.8 km + 23.25 km - 51.0 km = 12.05 km
Next, we can calculate the total time taken by adding the time spent in each direction:
Total time = Time east + Time north + Time west = 36.0 min + 30.0 min + 45.0 min = 111.0 min
Finally, we can calculate the average velocity of the train during this run by dividing the total displacement by the total time taken:
Average velocity = Total displacement / Total time = 12.05 km / (111.0 min / 60 min/h) ≈ 6.51 km/h
Therefore, the average velocity of the train during this run is approximately 6.51 km/h.