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

Young & Freedman Calc 14th Edition

Ch 03: Motion in Two or Three Dimensions

Problem 8a

Chapter 3, Problem 8a

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?

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Identify the given velocity function of the car: $ \mathbf{v}(t) = [5.00 \text{ m/s} - (0.0180 \text{ m/s}^3)t^2]\hat{i} + [2.00 \text{ m/s} + (0.550 \text{ m/s}^2)t]\hat{j} $.

Understand that the velocity function is composed of two components: $ v_x(t) = 5.00 \text{ m/s} - (0.0180 \text{ m/s}^3)t^2 $ and $ v_y(t) = 2.00 \text{ m/s} + (0.550 \text{ m/s}^2)t $.

To find the acceleration components, differentiate the velocity components with respect to time. The acceleration is the derivative of velocity.

Calculate the x-component of acceleration: $ a_x(t) = \frac{d}{dt}[5.00 \text{ m/s} - (0.0180 \text{ m/s}^3)t^2] $.

Calculate the y-component of acceleration: $ a_y(t) = \frac{d}{dt}[2.00 \text{ m/s} + (0.550 \text{ m/s}^2)t] $.

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Key Concepts

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

Velocity as a Function of Time

Velocity as a function of time describes how the speed and direction of an object change over time. In this problem, the velocity is given as a vector with components in the x and y directions, each expressed as a function of time. Understanding this concept is crucial for determining how the car's motion evolves in both directions.

Differentiation

Differentiation is a mathematical process used to find the rate at which a quantity changes. In physics, it is often used to derive acceleration from velocity. For this problem, differentiating the velocity functions with respect to time will yield the acceleration components ax(t) and ay(t), which describe how the car's velocity changes in the x and y directions.

Vector Components

Vector components break down a vector into parts that align with the coordinate axes, typically x and y. Each component represents the influence of the vector in that direction. In this problem, the velocity vector is split into x and y components, allowing us to analyze the car's motion separately along each axis, which is essential for calculating ax(t) and ay(t).

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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 $t equals 8.00 s$ ?

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

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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.

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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.

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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.

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