91Ó°ÊÓ

Displacement refers to the change in position of an object from its starting point to its ending point in a straight line. Unlike distance, which only considers the length of the path traveled, displacement takes into account the direction of travel.
In our exercise, the balloon moves in multiple dimensions: north, east, and upward. To find the displacement, we create a displacement vector, which combines these three directions.

• North: 21.5 km
• East: 9.7 km
• Upward: 2.88 km

By using these components, we treat the balloon’s movement as a single vector represented as \( \vec{d} = (21.5, 9.70, 2.88) \text{ km} \). This vector allows us to work with the balloon's movement in a structured way, making further calculations straightforward.

The magnitude of a vector tells us how long or large the vector is. For a displacement vector, the magnitude provides the overall distance between the starting point and the endpoint, regardless of the path taken.
To calculate magnitude in a three-dimensional space, we use the Pythagorean theorem extended to three dimensions, given by the formula: \[ |\vec{d}| = \sqrt{(21.5)^2 + (9.7)^2 + (2.88)^2} \]This involves squaring each component (north, east, and upward displacements), summing them up, and then taking the square root.
In this exercise, the calculation results in a magnitude of approximately 23.77 km. This distance represents the straight-line connection between the point where the balloon started and where it ended up after drifting.

Understanding vectors and how to calculate with them is crucial in physics. A vector carries both magnitude and direction, making it different from a simple scalar, which has only magnitude.
To find the average velocity using vectors, we use the total displacement and divide it by the time duration of the movement. The calculation for average velocity \( v_{avg} \) is:\[ v_{avg} = \frac{|\vec{d}|}{t} = \frac{23.77 \text{ km}}{3.50 \text{ h}} \approx 6.79 \text{ km/h} \]This gives us the speed and direction collectively as the average velocity of the balloon during its floating period. Vectors make sure that the directional aspect of velocity is always taken into account, giving a more complete understanding of motion.

Calculating the angle with the horizontal helps us determine how much the vertical movement influences the overall trajectory. In this case, it tells us how much the balloon ascended relative to its horizontal journey.
To find the angle \( \theta \), we use the tangent function, which is the ratio of the vertical displacement to the horizontal component magnitude. The formula used is:\[ \tan \theta = \frac{2.88}{23.58} \]This gives us the angle \( \theta \) with the horizontal after calculating the arctan (also known as the inverse tangent):\[ \theta = \tan^{-1} \left( \frac{2.88}{23.58} \right) \approx 6.97^\circ \]This angle indicates that the balloon had a slight ascend compared to its extensive horizontal drift. Such calculations are significant in understanding the nature of three-dimensional movement in space.