A train moving at a constant speed of 100km/h travels east for 40min, then 30o east of north for 20 min, and finally west for 30min. What is the train's average velocity for the trip?
The answer is 30 km/h (E 40 N) but I keep getting 66.6 km/h (E 30 N) help please!
To find the average velocity, we need to consider both the magnitude and direction of the displacement.
Let's break down the train's motion into three separate parts:
1. Eastward motion for 40 minutes at a constant speed of 100 km/h:
Distance = Speed × Time
Distance = (100 km/h) × (40/60 h) = 66.7 km
2. Northeastward motion at 30° east of north for 20 minutes:
To find the displacement, we need to calculate the horizontal and vertical components separately.
Horizontal displacement = Distance × cos(theta)
Vertical displacement = Distance × sin(theta)
Distance = Speed × Time
Distance = (100 km/h) × (20/60 h) = 33.3 km
Horizontal displacement = (33.3 km) × cos(30°) = 28.9 km
Vertical displacement = (33.3 km) × sin(30°) = 16.7 km
3. Westward motion for 30 minutes at a constant speed of 100 km/h:
Distance = Speed × Time
Distance = (100 km/h) × (30/60 h) = 50 km
Now, let's calculate the total displacement of the train:
Horizontal displacement = Eastward displacement - Westward displacement
Horizontal displacement = 66.7 km - 50 km = 16.7 km east
Vertical displacement = Northeastward displacement
Vertical displacement = 16.7 km north
Using the Pythagorean theorem, we can find the magnitude of the displacement:
Magnitude of displacement = sqrt((horizontal displacement)^2 + (vertical displacement)^2)
Magnitude of displacement = sqrt((16.7 km)^2 + (16.7 km)^2) = 23.6 km
Finally, we can find the average velocity by dividing the magnitude of displacement by the total time:
Average velocity = Magnitude of displacement / Total time
Average velocity = 23.6 km / ((40/60 h) + (20/60 h) + (30/60 h)) = 30 km/h (east 40° north)
Therefore, the correct average velocity for the trip is 30 km/h (east 40° north).
To find the train's average velocity for the trip, we can break it down into two components: the eastward component and the northward component.
Step 1: Convert the time durations into hours.
- 40 minutes is equal to 40/60 = 2/3 hours.
- 20 minutes is equal to 20/60 = 1/3 hours.
- 30 minutes is equal to 30/60 = 1/2 hours.
Step 2: Calculate the total eastward displacement.
The train travels east for 40 minutes at a speed of 100 km/h. So, the eastward displacement is given by:
Distance = Speed x Time = 100 km/h x (2/3) hours = 200/3 km.
Step 3: Calculate the northward displacement.
The train travels at 30° east of north for 20 minutes at a speed of 100 km/h. To calculate the northward displacement, we need to find the component of the speed that is pointing north. This can be done using trigonometry.
Northward speed = Speed x sin(theta) = 100 km/h x sin(30°) = 100 km/h x 1/2 = 50 km/h.
Now, we can find the northward displacement:
Distance = Speed x Time = 50 km/h x (1/3) hours = 50/3 km.
Step 4: Calculate the westward displacement.
The train travels west for 30 minutes at a speed of 100 km/h. So, the westward displacement is given by:
Distance = Speed x Time = 100 km/h x (1/2) hours = 50 km.
Step 5: Calculate the total displacement (resultant vector).
To find the total displacement, we need to add the eastward, northward, and westward displacements together. Since the eastward and westward displacements are in opposite directions, we subtract them:
Total eastward displacement = 200/3 km.
Total northward displacement = 50/3 km.
Total westward displacement = -50 km (negative sign indicates westward direction).
Total displacement = Total eastward displacement + Total northward displacement + Total westward displacement
= (200/3 km) + (50/3 km) + (-50 km)
= 250/3 km - 50 km
= 250/3 km - 150/3 km
= 100/3 km.
Step 6: Calculate the average velocity.
Average velocity = Total displacement / Total time taken.
Total time taken = (2/3) hours + (1/3) hours + (1/2) hours = 7/6 hours.
Average velocity = (100/3 km) / (7/6 hours)
= (100/3 km) x (6/7 hours)
= 600/21 km/h
= 28.57 km/h (approximately).
Therefore, the correct average velocity for the trip is approximately 28.57 km/h (E 30 N), not 30 km/h (E 40 N), as you mentioned.