A train with a constant speed of 60km/h goes east for 40 minutes. Then it goes

45◦ north-east for 20 minutes. And finally it goes west for 50 minutes. What
is the average velocity of the train?

Draw a diagram, where the train starts at (0,0). The speed makes it easy to convert from minutes to km. The final location is

(40,0)+(10√2,10√2)+(-50,0) = (10√2-10,10√2)

So, just convert that location to distance and bearing from (0,0).

Displacement = V*T1 + V*T2 + V*T3.

Disp. = 60*(40/60) + 60*(20/60)[45o] + (-60(50/60),
Disp. = 40 + (14.14+14.14i) + (-50) = 4.14 + 14.14i = 14.73km[73.7o].

T = T1+T2+T3 = 40 + 20 + 50 = 110 Min. = 1.833 Hours.

V = Disp./T = 14.73km[73.7o]/1.833 = 8.03km/h[73.7o].

To find the average velocity, we need to calculate the total displacement and divide it by the total time taken.

1. Convert the time durations into hours:
- 40 minutes = 40/60 = 2/3 hour
- 20 minutes = 20/60 = 1/3 hour
- 50 minutes = 50/60 = 5/6 hour

2. Calculate the distance traveled in each leg of the journey:
- Eastward: The train travels for 2/3 hour at a constant speed of 60 km/h, so the distance covered is (2/3) * 60 = 40 km.
- North-eastward: The train travels for 1/3 hour at a constant speed of 60 km/h, so the distance covered is (1/3) * 60 = 20 km.
- Westward: The train travels for 5/6 hour at a constant speed of 60 km/h, so the distance covered is (5/6) * 60 = 50 km.

3. Calculate the total displacement:
To find the total displacement, we need to consider the direction and magnitude of each leg of the journey. We can represent the eastward direction as positive x-axis and the northward direction as positive y-axis.

- Eastward: The displacement is 40 km to the right (positive x-direction).
- North-eastward: The displacement is 20 km in the northeast direction, which has components of 20*cos(45°) = 20*(√2 / 2) = 10√2 km in the positive x-direction and 20*sin(45°) = 20*(√2 / 2) = 10√2 km in the positive y-direction.
- Westward: The displacement is 50 km to the left (negative x-direction).

To find the net displacement, we need to add the displacements in each direction:
- Displacement in the x-direction = 40 km + 10√2 km - 50 km = 10√2 - 10 km.
- Displacement in the y-direction = 10√2 km.

4. Calculate the total time:
The total time taken is the sum of the time taken in each leg of the journey: 2/3 hour + 1/3 hour + 5/6 hour = (4 + 2 + 5) / 6 = 11/6 hour.

5. Calculate the average velocity:
Average velocity = Total displacement / Total time taken
Average velocity = (10√2 - 10) km / (11/6) hour
Average velocity = (10√2 - 10) * (6/11) km/h

Therefore, the average velocity of the train is approximately (10√2 - 10) * (6/11) km/h.

To find the average velocity of the train, we need to combine the individual velocities of each segment and then divide by the total time taken.

Let's break down each segment's velocity:

1. Eastward Segment: The train travels at a constant speed of 60 km/h for 40 minutes. As time is in minutes, we need to convert it to hours to match the units of the speed. Since there are 60 minutes in an hour, 40 minutes is equivalent to 40/60 = 2/3 hours. Therefore, the distance traveled in this segment is (60 km/h) * (2/3 hours) = 40 km.

2. North-East Segment: The train goes at a 45° angle, which means its motion can be split into a northward component and an eastward component. Since the northward and eastward components are perpendicular, we can treat them separately using trigonometry.

The distance traveled in the northward direction can be found by multiplying the speed (60 km/h) by the time (20 minutes) and then multiplying by the sine of the angle. Since sine(45°) = √2/2, the northward distance is (60 km/h) * (20/60 hours) * (√2/2) = 10√2 km.

Similarly, the distance traveled in the eastward direction can be found by multiplying the speed (60 km/h) by the time (20 minutes) and then multiplying by the cosine of the angle. Since cosine(45°) = √2/2, the eastward distance is (60 km/h) * (20/60 hours) * (√2/2) = 10√2 km.

3. Westward Segment: The train travels westward at a constant speed of 60 km/h for 50 minutes. As before, we need to convert the time to hours. 50 minutes is equivalent to 50/60 = 5/6 hours. Therefore, the distance traveled in this segment is (60 km/h) * (5/6 hours) = 50 km.

Now, let's calculate the total displacement by considering the distance traveled in the x and y directions. The net displacement in the x-direction is 40 km westward (negative), and the net displacement in the y-direction is 10√2 km northward.

To find the total displacement, we use the Pythagorean theorem. The magnitude of the displacement is the square root of the sum of the squares of the individual displacements:

|Δd| = √((40 km)^2 + (10√2 km)^2)

Finally, we need to calculate the average velocity by dividing the total displacement magnitude by the total time taken:

Average Velocity = |Δd| / Total Time

Total Time = 40 minutes + 20 minutes + 50 minutes = 110 minutes.

Total Time converted to hours = 110/60 hours.

Now we can substitute the values into the formula to find the average velocity:

Average Velocity = (√((40 km)^2 + (10√2 km)^2)) / (110/60 hours)