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. (c) Calculate the magnitude and direction of the bird's velocity and acceleration at t = 2.0 s.

Verified step by step guidance

1. First, determine the velocity components of the bird. The velocity in the x-direction, v_x(t), is the derivative of x(t) with respect to time t. Since x(t) = αt, the derivative is v_x(t) = α. For the given α = 2.4 m/s, v_x(t) = 2.4 m/s.
2. Next, find the velocity in the y-direction, v_y(t), by differentiating y(t) with respect to time t. Given y(t) = 3.0 m − βt², the derivative is v_y(t) = -2βt. Substitute β = 1.2 m/s² to get v_y(t) = -2(1.2)t = -2.4t m/s.
3. Calculate the velocity components at t = 2.0 s. For v_x, it remains constant at 2.4 m/s. For v_y, substitute t = 2.0 s into v_y(t) = -2.4t to get v_y(2.0) = -2.4(2.0) = -4.8 m/s.
4. Determine the magnitude of the velocity vector using the Pythagorean theorem: |v| = √(v_x² + v_y²). Substitute v_x = 2.4 m/s and v_y = -4.8 m/s into the equation.
5. Find the direction of the velocity vector by calculating the angle θ with respect to the positive x-axis using the tangent function: θ = arctan(v_y/v_x). Substitute v_y = -4.8 m/s and v_x = 2.4 m/s into the equation.

Key Concepts

Kinematics in Two Dimensions

Kinematics involves the study of motion without considering the forces that cause it. In two dimensions, motion is described using position, velocity, and acceleration vectors. The position of an object is given by its coordinates, and its velocity and acceleration are the first and second derivatives of the position with respect to time, respectively.

Velocity and Acceleration Vectors

The velocity vector describes the rate of change of position with time and has both magnitude and direction. It is calculated by differentiating the position functions with respect to time. Similarly, the acceleration vector, which describes the rate of change of velocity, is found by differentiating the velocity functions. These vectors are crucial for understanding the motion of objects in the xy-plane.

Magnitude and Direction of Vectors

The magnitude of a vector in two dimensions is found using the Pythagorean theorem, combining its components. The direction is determined by calculating the angle the vector makes with a reference axis, typically using the arctangent function. These calculations are essential for fully describing the velocity and acceleration of an object in motion.