A train moving at a constant speed of 68.0 km/h moves east for 33.0 min, then in a direction 40.0° east of due north for 30.0 min, and then west for 59.0 min. What is the average velocity of the train during this run?

Magnitude
km/h
Direction
° (counterclockwise east)

East motion

x = 68 *33/60 * 68 sin 40 *30/60 - 68 *59/60
= 68* -.112 = -7.61 km east
North motion
68 * cos 40 * 33/60 = 28.7 km north
tan angle west of north = 7.61/28.7
so
angle west of north = 14.9
angle from East = 90 + 14.9 = 104.9

To find the average velocity of the train during this run, we need to calculate its total displacement and the total time taken.

First, let's find the displacement in the east direction. The train moves at a constant speed of 68.0 km/h for 33.0 minutes, which is (33/60) hours. So, the displacement in the east direction is:

Displacement east = Speed east * Time east
Displacement east = 68.0 km/h * (33/60) h

Next, let's find the displacement in the north-east direction. The train moves at a constant speed of 68.0 km/h in a direction 40.0° east of due north for 30.0 minutes, which is (30/60) hours. We can break this velocity into its east and north components:

Velocity east = Speed northeast * cos(angle)
Velocity north = Speed northeast * sin(angle)

Velocity east = 68.0 km/h * cos(40.0°)
Velocity north = 68.0 km/h * sin(40.0°)

Now, let's find the displacement in the west direction. The train moves at a constant speed of 68.0 km/h for 59.0 minutes, which is (59/60) hours. So, the displacement in the west direction is:

Displacement west = Speed west * Time west
Displacement west = 68.0 km/h * (59/60) h

To find the total displacement, we need to add up the east, north-east, and west displacements:

Total Displacement east = Displacement east + Velocity east * Time northeast
Total Displacement north = Velocity north * Time northeast
Total Displacement west = Displacement west

Total Displacement east = 68.0 km/h * (33/60) h + (68.0 km/h * cos(40.0°)) * (30/60) h
Total Displacement north = (68.0 km/h * sin(40.0°)) * (30/60) h
Total Displacement west = 68.0 km/h * (59/60) h

Finally, we can calculate the total displacement using the Pythagorean theorem:

Total Displacement = sqrt((Total Displacement east)^2 + (Total Displacement north)^2 + (Total Displacement west)^2)

Now, we need to calculate the total time taken:

Total Time = Time east + Time northeast + Time west

Average Velocity = Total Displacement / Total Time

Calculate the magnitude and direction of the average velocity based on the calculated values.