Question

A plane flies \(410 \mathrm{mi}\) east from city \(A\) to city \(B\) in \(45 \mathrm{~min}\) and then \(820 \mathrm{mi}\) south from city \(B\) to city \(C\) in 1 h \(30 \mathrm{~min}\). (a) What are the magnitude and direction of the displacement vector that represents the total trip? What are \((b)\) the average velocity vector and \((c)\) the average speed for the trip?

Short answer

The total displacement is \(900\) miles, \(63.43°\) south of east, the average velocity is \(514.29\) miles/hr in the same direction, and the average speed is \(702.86\) miles/hr.

Step-by-step solution

Calculate the total displacement

The total displacement is calculated by vector addition. Because these vectors are perpendicular (east and south are perpendicular directions), they form a right-angled triangle. The displacement to the east can be represented by a vector \([410,0]\) and the displacement to the south by a vector \([0,-820]\). Adding these vectors gives \([410,-820]\). Using Pythagoras' theorem (\( \sqrt{x^2 + y^2} \)), we get the magnitude of displacement as \( \sqrt{410^2 + (-820)^2} = 900 \) miles. To find the direction, calculate the inverse tangent of the absolute value of the vertical displacement divided by the horizontal displacement. So \( \theta = \tan^{-1}(\frac{-820}{410}) = -63.43° \). Bearing in mind that south of east is a negative angle, our final angle is \(63.43°\) south of east.

Calculate the average velocity

The average velocity is calculated by dividing the displacement vector by the total time for the journey. First, convert all the times into the same units to calculate the total travel time, which is \(45 min + 1.5h = 1.75 h\). The average velocity vector is the total displacement vector \([410, -820]\) divided by the total time \(1.75h\), thus \([234.3, -468.6]\) miles/hr. In other words, the velocity is \(900/1.75 = 514.29\) miles/hr. The direction remains \(63.43°\) south of east as the average velocity vector and displacement vector always have the same direction.

Calculate the average speed

The average speed is calculated by dividing total distance travelled by the total time for the journey. In this case, the total distance travelled is \(410 + 820 = 1230 \) miles, and the total time is still \(1.75 h\). Dividing the total distance by the total time, the average speed works out to \(1230 / 1.75 = 702.86\) miles/hr.

Key concepts

Average Velocity

To understand average velocity, it's important to know that it is a vector quantity. This means that it has both a magnitude and a direction. In our exercise, we look at the plane's total trip which involved traveling east and then south. The average velocity is calculated using the total displacement of the plane divided by the total time taken for the journey.

• Total Displacement Vector: The displacement is represented as a vector. Since the movements east and south form perpendicular directions, our displacement forms a right-angled triangle.
• Time: We had to convert everything into hours for a uniform measurement. So, we get a total journey time of 1.75 hours.
• Calculation: We divided the displacement vector \[ [410, -820] \] by 1.75 hours to get the average velocity vector \[ [234.3, -468.6] \] miles/hr.

Average velocity also takes direction into account, which in this case is 63.43° south of east, aligning with the direction of displacement.

Average Speed

Average speed is a bit different from average velocity because it is a scalar quantity. This means it doesn't have a direction, just a magnitude. It simply tells us how fast the plane was moving overall during its journey. Unlike average velocity, which considers only the start and end points, average speed uses the total path traveled.

• Total Distance: The plane covered a total of 1230 miles (410 miles east + 820 miles south).
• Total Time: Just like before, the total travel time was 1.75 hours.
• Calculation: Average speed is calculated by dividing the total distance by the total time: \[1230 \, \text{miles}/1.75 \, \text{hours} = 702.86 \, \text{miles per hour}\]

Because it's a scalar, we don't specify a direction for average speed, just how fast the plane moved on average.

Right-Angled Triangle

The right-angled triangle is a fundamental concept in geometry and plays a crucial role in this exercise when calculating the plane's total displacement. When two movements are perpendicular to each other, like traveling east and then south, they naturally form the two shorter sides of a right-angled triangle.

• Pythagoras' Theorem: This theorem helps us find the length of the hypotenuse (total displacement in this context). The formula is \[c = \sqrt{a^2 + b^2}\], where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the hypotenuse.
• Vectors As Triangle Sides: In the problem, the east and south displacements \[ [410, 0] \] and \[ [0, -820] \] are the legs of the triangle. Solving for the hypotenuse gives us the displacement's magnitude of 900 miles.
• Direction Using Angles: The direction angle relative to east is calculated with \[ \theta = \tan^{-1} \left(\frac{-820}{410}\right) \] which helps in establishing the plane's direction as 63.43° south of east.

This concept helps visualize motion and is crucial for determining the resultant path in two-dimensional space.