A train travelling at a constant speed of 100 km/h travels west for 45 minutes, northwest for 30 minutes and finally north for 15 minutes. What is the trains average velocity for the trip?
Disp. = 100*(45/60)[180o] + 100*(30/60)[135o] + 100*(15/60)[90o].
Disp. = 100*0.75[180o] + 100*0.5[135] + 100*0.25[90o],
Disp. = -75 + 50[135] + 25[90o],
Disp. = (-75-50*Cos135) + (50*sin135+25)i = -110.4 + 60.4i = 126km[-28.7o] = 126km[28.7o N. of W.] = 126km[151.3o CCW).
V = Disp./t = 126km[151.3o]/(0.75+0.5+0.25) = 84km/h[151.3o].
To find the average velocity of the train for the entire trip, we need to calculate the total displacement and divide it by the total time taken.
First, we need to determine the total displacement. Displacement is a vector quantity that represents the change in position.
Let's break down the train's motion into its components:
1. West: The train travels at a constant speed of 100 km/h for 45 minutes. We can calculate the displacement in the west direction using the formula: Displacement = Velocity × Time. Since the train is moving west, the velocity is -100 km/h (negative because it is in the opposite direction) and the time is 45 minutes. Therefore, the displacement in the west direction is -100 km/h × (45/60) h = -75 km.
2. Northwest: The train travels in this direction for 30 minutes. Since this is not purely west, we need to calculate the displacement in the northwest direction. Since we don't have the exact angle, let's assume it is at a 45-degree angle. The velocity can be decomposed into two components: 100 km/h × cos(45°) in the west direction and 100 km/h × sin(45°) in the north direction. The time is 30 minutes. Therefore, the displacement in the northwest direction is: (100 km/h × cos(45°)) × (30/60) h = 50√2 km.
3. North: The train travels in this direction for 15 minutes. The velocity is purely in the north direction, which means there is no westward displacement. The velocity is 0 km/h in the west direction, and the time is 15 minutes. Therefore, the displacement in the north direction is 100 km/h × (15/60) h = 25 km.
Now, let's calculate the total displacement:
Total displacement = Displacement in the west direction + Displacement in the northwest direction + Displacement in the north direction
= -75 km + 50√2 km + 25 km
To calculate the average velocity, divide the total displacement by the total time taken:
Total time taken = 45 minutes + 30 minutes + 15 minutes = 90 minutes = 1.5 hours
Average velocity = Total displacement / Total time taken
= (-75 km + 50√2 km + 25 km) / 1.5 hours
Now you can calculate the average velocity using the given values.