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. (b) Calculate the velocity and acceleration vectors of the bird as functions of time.

Verified step by step guidance

1. First, identify the expressions for the position coordinates x(t) and y(t). From the problem, we have x(t) = 2.4t and y(t) = 3.0 - 1.2t^2.
2. Next, calculate the velocity components in the x and y directions. The velocity is the first derivative of the position with respect to time. Thus, differentiate x(t) and y(t) with respect to t to find the velocity components vx(t) and vy(t).
3. For the x-component of velocity, differentiate x(t) = 2.4t with respect to t to get vx(t).
4. For the y-component of velocity, differentiate y(t) = 3.0 - 1.2t^2 with respect to t to get vy(t).
5. Finally, calculate the acceleration components in the x and y directions. The acceleration is the derivative of the velocity with respect to time. Thus, differentiate vx(t) and vy(t) with respect to t to find the acceleration components ax(t) and ay(t).

Key Concepts

Position Functions

Position functions describe the location of an object in space as a function of time. In this case, the bird's position is given by x(t) = αt for horizontal movement and y(t) = 3.0 m − βt² for vertical movement. Understanding these functions is crucial for determining how the bird's position changes over time.

Velocity Vector

The velocity vector represents the rate of change of position with respect to time. It is calculated by taking the derivative of the position functions with respect to time. For the bird, the velocity vector will have components derived from the derivatives of x(t) and y(t), indicating both horizontal and vertical speeds.

Acceleration Vector

The acceleration vector indicates the rate of change of velocity with respect to time. It is found by differentiating the velocity vector. In this scenario, the acceleration will reveal how the bird's speed and direction change over time, particularly influenced by the quadratic term in the y-position function.