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 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

All textbooks

Young & Freedman Calc 14th 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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

11m

Was this helpful?

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.

Recommended video:

Guided course

06:44

Adding 3 Vectors in Unit Vector Notation

Related Practice

Textbook Question

A remote-controlled car is moving in a vacant parking lot. The velocity of the car as a function of time is given by $\vec{v}\) = $\left$[ 5.00~$\mathrm{m/s}$ - (0.0180~$\mathrm{m/s^3}$)t^2 $\right$] $\hat{i}$ + $\left$[ 2.00~$\mathrm{m/s}$ + (0.550~$\mathrm{m/s^2}$)t $\right$] \(\hat{j}$ . What are the magnitude and direction of the car's velocity at $t equals 8.00 s$ ? (b) What are the magnitude and direction of the car's acceleration at $t equals 8.00 s$ ?

views

Textbook Question

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.

views

Textbook Question

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.

views

Textbook Question

A remote-controlled car is moving in a vacant parking lot. The velocity of the car as a function of time is given by $\vec{v}\) = $\left$[ 5.00~$\mathrm{m/s}$ - (0.0180~$\mathrm{m/s^3}$)t^2 $\right$] $\hat{i}$ + $\left$[ 2.00~$\mathrm{m/s}$ + (0.550~$\mathrm{m/s^2}$)t $\right$] \(\hat{j}$ . What are $a_{x} (t)$ and $a_{y} (t)$ , the $x$ - and $y$ - components of the car's velocity as functions of time?

views

Textbook Question

A physics book slides off a horizontal tabletop with a speed of 1.10 m/s. It strikes the floor in 0.480 s. Ignore air resistance. Find the height of the tabletop above the floor.

views

Textbook Question

A dog running in an open field has components of velocity v_x = 2.6 m/s and v_y = −1.8 m/s at t1 = 10.0 s. For the time interval from t₁ = 10.0 s to t₂ = 20.0 s, the average acceleration of the dog has magnitude 0.45 m/s² and direction 31.0° measured from the +x–axis toward the +y–axis. At t₂ = 20.0 s, what are the x- and y-components of the dog's velocity?

views