91Ó°ÊÓ

When dealing with motion in physics, particularly in two dimensions, understanding the concept of velocity components is crucial. Velocity is a vector, meaning it has both a magnitude and a direction. To analyze its effects thoroughly, we often break it down into components – typically along the x (horizontal) and y (vertical) axes. In our exercise, the bird initially has a velocity of 21.1 m/s entirely in the east, which means all of its velocity is along the horizontal axis.
After a second, although the magnitude of the velocity remains the same, the direction changes by 7° north of east. To find out how this affects the motion, we decompose this velocity into two components:

• Horizontal (eastward) component: \(v_{fx} = 21.1 \cos(7^\circ) \)
• Vertical (northward) component: \(v_{fy} = 21.1 \sin(7^\circ) \)

These components tell us how much of the bird's velocity is along each axis, assisting in further calculations like acceleration.

Acceleration is all about how quickly an object's velocity changes over time. In this problem, since we assume constant acceleration, the bird’s change in velocity over the one-second interval directly relates to acceleration.
After calculating the change in velocity components, the acceleration components can be calculated as follows:

• Horizontal acceleration \(a_x = \frac{\Delta v_x}{\Delta t}\)
• Vertical acceleration \(a_y = \frac{\Delta v_y}{\Delta t}\)

Given \(\Delta t = 1\, \text{s}\), the changes in velocity components are simply equivalent to the acceleration components. Thus, the change from initial to final velocity dictates how the bird accelerates in both horizontal and vertical directions.

Kinematics is the study of motion without considering its causes. It focuses on the description of motion, which here includes the bird's velocity and acceleration.
In this scenario, the motion involves the initial and final velocities and how these change over time due to acceleration. By using kinematic equations, we can clearly determine the relationships between distance, velocity, acceleration, and time. The primary kinematic equation used here is:

• \(\Delta v = a \cdot \Delta t\)

Understanding this equation allows us to connect the concepts of velocity change and acceleration, providing a complete picture of how the bird's flight path changes in the given time period.

Trigonometry plays an essential role in physics, especially in resolving vector components. By using trigonometric functions such as sine and cosine, we can break down a vector into perpendicular components that are easier to work with.
In this exercise, the angle information provided (7° north of east) was critical in calculating the final velocity components. Here's how trigonometry helped:

• To find the horizontal component \(v_{fx} = 21.1 \cos(7^\circ)\)
• To find the vertical component \(v_{fy} = 21.1 \sin(7^\circ)\)

Using these trigonometric identities, we converted directional velocity into components that could be analyzed algebraically, aiding in the subsequent calculation of acceleration. Without trigonometry, resolving such vectors would be highly complex.