The coordinates of a bird flying in the xy-plane are given by x(t) = αt and y(t) = 3.0 m − βt2, where α = 2.4 m/s and β = 1.2 m/s2. (a) Sketch the path of the bird between t = 0 and t = 2.0 s.

Verified step by step guidance

1. Understand the equations of motion: The bird's position in the xy-plane is described by two equations: x(t) = αt and y(t) = 3.0 m − βt², where α = 2.4 m/s and β = 1.2 m/s². These equations give the x and y coordinates of the bird at any time t.
2. Determine the range of time: We need to sketch the path of the bird between t = 0 and t = 2.0 s. This means we will calculate the bird's position at several points within this time interval.
3. Calculate the x and y coordinates at key time points: Start by calculating the coordinates at t = 0, t = 1.0 s, and t = 2.0 s. For example, at t = 0, x(0) = α * 0 = 0 and y(0) = 3.0 m − β * 0² = 3.0 m. Repeat this for t = 1.0 s and t = 2.0 s.
4. Plot the points on the xy-plane: Using the coordinates calculated, plot the points on a graph with the x-axis representing x(t) and the y-axis representing y(t). Connect these points to visualize the path of the bird.
5. Analyze the path: Observe the shape of the path. The x-coordinate increases linearly with time, while the y-coordinate decreases quadratically, indicating the bird is moving forward while descending in a parabolic trajectory.

Key Concepts

Parametric Equations

Parametric equations express the coordinates of a point in terms of a parameter, often time. In this problem, x(t) = αt and y(t) = 3.0 m − βt² describe the bird's position as functions of time, allowing us to determine its trajectory in the xy-plane by varying t.

Projectile Motion

Projectile motion involves objects moving in two dimensions under the influence of gravity. The bird's y-coordinate follows a quadratic equation, typical of vertical motion under constant acceleration, while the x-coordinate changes linearly, indicating uniform horizontal motion.

Graphing Techniques

Graphing techniques are essential for visualizing mathematical relationships. To sketch the bird's path, plot x(t) and y(t) for values of t from 0 to 2.0 s, showing how the bird's position changes over time, and illustrating the combined effects of linear and quadratic motion.