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

Ch 03: Motion in Two or Three Dimensions

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 03: Motion in Two or Three Dimensions

Problem 7c

Chapter 3, Problem 7c

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

Verified step by step guidance

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. Similarly, the velocity in the y-direction, v_y(t), is the derivative of y(t) with respect to time t. Given y(t) = 3.0 m − βt², the derivative is v_y(t) = -2βt.

Next, calculate the velocity components at t = 2.0 s. Substitute t = 2.0 s into the expressions for v_x(t) and v_y(t). For v_x(t), it remains constant as α = 2.4 m/s. For v_y(t), substitute t = 2.0 s into v_y(t) = -2βt to find v_y(2.0 s).

To find the magnitude of the bird's velocity at t = 2.0 s, use the Pythagorean theorem. The magnitude of the velocity vector v is given by v = √(v_x² + v_y²). Substitute the values of v_x and v_y at t = 2.0 s into this formula.

Determine the direction of the bird's velocity at t = 2.0 s. The direction θ can be found using the tangent function: θ = arctan(v_y/v_x). Substitute the values of v_y and v_x at t = 2.0 s into this formula to find the angle θ.

Finally, calculate the acceleration components. The acceleration in the x-direction, a_x(t), is the derivative of v_x(t) with respect to time, which is zero since v_x(t) is constant. The acceleration in the y-direction, a_y(t), is the derivative of v_y(t) with respect to time, which is a_y(t) = -2β. Calculate a_y(t) at t = 2.0 s using β = 1.2 m/s².

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Velocity in Two Dimensions

Velocity in two dimensions involves both the x and y components, which can be derived from the derivatives of the position functions with respect to time. For the bird's motion, the velocity components are vx = dx/dt = α and vy = dy/dt = -2βt. The magnitude of the velocity is found using the Pythagorean theorem: v = √(vx² + vy²).

Acceleration in Two Dimensions

Acceleration in two dimensions also has x and y components, obtained from the second derivatives of the position functions. For the bird, the acceleration components are ax = d²x/dt² = 0 and ay = d²y/dt² = -2β. The magnitude of the acceleration is calculated similarly to velocity: a = √(ax² + ay²).

Direction of a Vector

The direction of a vector in the xy-plane is determined by the angle it makes with the positive x-axis, calculated using trigonometry. For velocity and acceleration, the angle θ can be found using tan(θ) = vy/vx for velocity and tan(θ) = ay/ax for acceleration. This angle provides insight into the vector's orientation relative to the coordinate axes.

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Textbook Question

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

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