Question

In 3 h 24 min, a balloon drifts $8.7\mathrm{km}$ north, $9.7\mathrm{km}$ east, and $2.9\mathrm{km}$ in elevation from its release point on the ground. Find (a) the magnitude of its average velocity and $(b)$ the angle its average velocity makes with the horizontal.

Short answer

The average velocity of the balloon is both 0.0017 km/s in magnitude and directed at an angle of 10.71 degrees from the horizontal.

Step-by-step solution

Calculation of displacement vector

The balloon moves 8.7 km North, 9.7 km East, and 2.9 km Upward from its release point. Hence, the displacement vector $\overrightarrow{d}$ is \{8.7 , 9.7 , 2.9\} km.

Conversion of time into proper unit

The time given is 3 hours 24 minutes. This needs to be converted into seconds in order to have a consistent set of units. $3h=3\times 3600s=10800s$, $24min=24\times 60s=1440s$, hence, total time $t=10800s+1440s=12240s$.

Calculation of magnitude of average velocity

To find the magnitude of the average velocity we use the formula for average velocity $v_{avg}$, which is the displacement over time. Therefore, magnitude of average velocity $v_{avg}$ is $\sqrt{({8.7}^2+{9.7}^2+{2.9}^2)}km/12240s=0.0017km/s$.

Calculation of direction of average velocity

To find the direction of the average velocity, we make use of the inverse tangent function (also known as arctan) applied to the height over the horizontal displacement. Let the direction be $\theta$. $\theta =atan(2.9/\sqrt{({8.7}^2+{9.7}^2)})=10.71degrees$.

Key concepts

Displacement Vector

When considering the movement of an object from one point to another, we describe its net movement using a displacement vector. In this scenario, the balloon’s journey can be expressed with a displacement vector, which includes its path 8.7 km north, 9.7 km east, and 2.9 km upward in elevation. This vector, represented as $\overrightarrow{d}=\{8.7,9.7,2.9\}\text{ km}$, provides a concise way to capture the total change in position of the balloon over time. Understanding this concept is crucial as it simplifies the complexity involved in describing motion in multiple dimensions by using this three-component system.

Magnitude Calculation

The calculation of magnitude for any vector gives us a scalar value that represents the vector’s size or length, without regard to its direction. In the case of the displacement vector $\overrightarrow{d}=\{8.7,9.7,2.9\}\text{ km}$, the magnitude can be found using the Euclidean distance formula:

• Square each component: ${8.7}^2,{9.7}^2,\text{ and }{2.9}^2$
• Sum these squares: ${8.7}^2+{9.7}^2+{2.9}^2$
• Take the square root of the sum to find the magnitude

This gives the magnitude of $\sqrt{({8.7}^2+{9.7}^2+{2.9}^2)}$ km. This magnitude is an essential part of determining other properties of movement, such as average velocity.

Direction of Motion

Determining the direction of motion for an object moving in three-dimensional space involves understanding the relationship between its horizontal and vertical movements. In this exercise, the balloon’s direction is assessed by focusing on the angle its velocity makes with the horizontal plane. To find this angle, we use the inverse tangent function. Specifically, we calculate:

• The horizontal component $\sqrt{({8.7}^2+{9.7}^2)}$ for the north-east movement
• Divide the elevation gain (vertical component) by this horizontal component for the tangent of the angle
• Utilize the arctan function to uncover the angle $\theta =\arctan \left(\frac{2.9}{\sqrt{({8.7}^2+{9.7}^2)}}\right)$

This results in an angle of approximately 10.71 degrees, informing us about how steeply the balloon ascends as it drifts away.

Vector Components

Breaking down a vector into its components is a fundamental tool in understanding complex movements in physics. Each vector in three-dimensional space consists of three parts: a component along the x-axis (east), y-axis (north), and z-axis (upward elevation). For the balloon example:

• Eastward movement: 9.7 km represents movement along the x-axis
• Northward movement: 8.7 km along the y-axis
• Upward movement: 2.9 km along the z-axis

These components can be visually represented in a three-dimensional coordinate system, facilitating the analysis of the object's trajectory and informing calculations like magnitude or directional angles. Understanding vector components is essential to breaking down and solving multidimensional motion problems.