A regional airline flight consists of two legs with an intermediate stop. The airplane flies 380 due north from airport A to airport B. From there, it flies 260 due east to its final destination at airport C.
Part A
What is the magnitude of the plane’s displacement from its starting point?
Part B
What is the direction of the plane’s displacement from its starting point?
Part C
If the first leg takes 50 and the second leg 45 , what is the magnitude of the average velocity for the trip?
Part D
What is the direction of the average velocity for the trip?
Part E
What is the average speed for the trip?
Part F
Why is the average speed not the same as the magnitude for the average velocity?
Part A: The magnitude of the plane's displacement from its starting point can be found using the Pythagorean theorem, since the plane flies in a right-angled triangle. So, the displacement is the square root of (380^2 + 260^2).
Part B: The direction of the plane's displacement from its starting point can be found by taking the inverse tangent of (260/380). But let's be honest, directions can be confusing, especially when you can't just ask a local squirrel for directions.
Part C: To find the magnitude of the average velocity for the trip, you need to divide the total displacement (found in Part A) by the total time it takes for the trip. Just remember, velocity is like your friend who always has a direction, unlike speed which is a bit directionless, like going in circles with a hula hoop.
Part D: The direction of the average velocity for the trip is the same as the direction of the plane's displacement from its starting point. Unless the pilot took a wrong turn at Albuquerque, then we're all in trouble.
Part E: The average speed for the trip can be found by dividing the total distance traveled (which is the sum of the distances for both legs) by the total time it takes for the trip. You know, speed is the rate at which the pilot is making hundreds of tiny decisions, like "Should I nap or order another cup of coffee?"
Part F: Ah, the difference between average speed and magnitude of average velocity is like trying to compare a cat and a unicorn. Average speed only cares about the total distance traveled over time, while magnitude of average velocity considers both distance and direction. So speed is like a fast-food order - it doesn't matter if you made a hundred wrong turns, as long as you make it on time. Velocity, on the other hand, is like a fancy chef's meal - every aspect, including direction, is taken into account.
Part A: To find the magnitude of the plane's displacement from its starting point, we need to calculate the total distance traveled. Since the first leg goes due north for 380 and the second leg goes due east for 260, we can use the Pythagorean theorem to find the total distance traveled (d):
d = sqrt((380)^2 + (260)^2)
= sqrt(144400 + 67600)
= sqrt(212000)
≈ 460.66
Therefore, the magnitude of the plane's displacement from its starting point is approximately 460.66.
Part B: To find the direction of the plane's displacement from its starting point, we can use trigonometry. Since the first leg is due north and the second leg is due east, we can calculate the angle (θ) using the inverse tangent function:
θ = atan((260)/(380))
= atan(0.6842)
≈ 36.87°
Therefore, the direction of the plane's displacement from its starting point is approximately 36.87° east of north.
Part C: The average velocity is defined as the total displacement divided by the total time taken. In this case, the total displacement is the same as the magnitude of the plane's displacement from its starting point (460.66), and the total time taken is the sum of the times for each leg (50 + 45 = 95). So we can calculate the average velocity (v) using the following formula:
v = (total displacement) / (total time)
= 460.66 / 95
≈ 4.85
Therefore, the magnitude of the average velocity for the trip is approximately 4.85 (units depend on the distance and time units used).
Part D: The direction of average velocity can be determined by combining the directions of the individual legs. Since the first leg is due north and the second leg is due east, we need to find the angle between due north and due east. This angle can be calculated as follows:
θ = atan((380)/(260))
= atan(1.4615)
≈ 55.16°
Therefore, the direction of the average velocity for the trip is approximately 55.16° east of north.
Part E: Average speed is defined as the total distance traveled divided by the total time taken. In this case, the total distance traveled is the same as the magnitude of the plane's displacement from its starting point (460.66), and the total time taken is the sum of the times for each leg (50 + 45 = 95). So we can calculate the average speed (s) using the following formula:
s = (total distance) / (total time)
= 460.66 / 95
≈ 4.85
Therefore, the average speed for the trip is approximately 4.85 (units depend on the distance and time units used).
Part F: The average speed is not the same as the magnitude of the average velocity because speed only considers the distance traveled without regard to direction, while velocity takes into account both distance and direction. In this case, the average speed is the same as the magnitude of the average velocity because the displacement is a straight line and there is no change in direction. However, in general, if the displacement involves changes in direction, the average speed will be greater than the magnitude of the average velocity.
Part A: To find the magnitude of the plane's displacement from its starting point, we can use the Pythagorean theorem. The first leg of the flight goes 380 due north, and the second leg goes 260 due east. These two legs form a right triangle. Using the Pythagorean theorem, we can calculate:
Magnitude of displacement = square root of (380^2 + 260^2)
Part B: To find the direction of the plane's displacement from its starting point, we can use trigonometry. Using the same right triangle with the legs of 380 and 260, we can find the angle using the inverse tangent function:
Direction of displacement = inverse tangent (260/380)
Part C: To find the magnitude of the average velocity for the trip, we need to calculate the total displacement and divide it by the total time taken. The total displacement is the same as the magnitude of the plane's displacement from its starting point calculated in Part A. The total time taken is the sum of the time spent on each leg of the flight.
Magnitude of average velocity = Magnitude of displacement / (50 + 45)
Part D: The direction of the average velocity for the trip is the same as the direction of the plane's displacement from its starting point calculated in Part B.
Part E: The average speed for the trip is calculated by dividing the total distance traveled by the total time taken. The total distance traveled is the sum of the distances traveled on each leg of the flight (380 + 260), and the total time taken is the sum of the time spent on each leg (50 + 45).
Average speed = (380 + 260) / (50 + 45)
Part F: The average speed is not the same as the magnitude of the average velocity because speed only measures the magnitude of the distance traveled per unit time, while velocity takes into account both the magnitude and the direction of the displacement per unit time. In this case, the plane's displacement is not a straight line distance from the starting point to the final destination, but rather a combination of north and east directions, resulting in a different magnitude when compared to the total distance traveled.