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Chapter 3: Problem 13

A vector is \(20.0 \mathrm{m}\) long and makes an angle of $60.0^{\circ} \mathrm{coun}-\( terclockwise from the \)y\( -axis (on the side of the \)-x$ -axis). What are the \(x\) - and \(y\) -components of this vector?

Short Answer

Expert verified
Answer: The x-component of the vector is -10.0 m, and the y-component is 17.32 m.

Step by step solution

01

Determine the reference angle

To find the x and y components, first, we need to find the reference angle from the -x-axis. Since the angle is measured counterclockwise from the y-axis, we know that: Reference angle = \((180^{\circ} - 60^{\circ})= 120^{\circ}\)
02

Find the x- and y-component using trigonometric functions

Trigonometric functions can be used to find the x- and y-components of the vector. For x-component, \( \textit{x} = \textit{magnitude} \times \textit{cos(reference angle)}\) For y-component, \( \textit{y} = \textit{magnitude} \times \textit{sin(reference angle)}\)
03

Calculate the x-component

Using the formula for x-component, and given magnitude, and reference angle: x = \(20.0 \ (\textit{m}) \times \textit{cos(120°)}\) x-component = \(20.0 \ (\textit{m}) \times -0.5\) (Note that the cos(120°) = -0.5) x-component = \(-10.0 \mathrm{m}\)
04

Calculate the y-component

Using the formula for y-component, and given magnitude, and reference angle: y = \(20.0 \ (\textit{m}) \times \textit{sin(120°)}\) y-component = \(20.0 \ (\textit{m}) \times 0.866\) (Note that the sin(120°) = 0.866) y-component = \(17.32 \mathrm{m}\)
05

Report the components

The x-component of the vector is -10.0 m and the y-component is 17.32 m.

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Most popular questions from this chapter

In each of these, the \(x\) - and \(y\) -components of a vector are given. Find the magnitude and direction of the vector. (a) $A_{x}=-5.0 \mathrm{m} / \mathrm{s}, A_{y}=+8.0 \mathrm{m} / \mathrm{s} .\( (b) \)B_{x}=+120 \mathrm{m}$ $B_{y}=-60.0 \mathrm{m} .(\mathrm{c}) C_{x}=-13.7 \mathrm{m} / \mathrm{s}, C_{y}=-8.8 \mathrm{m} / \mathrm{s} .(\mathrm{d}) D_{x}=$ \(2.3 \mathrm{m} / \mathrm{s}^{2}, D_{y}=6.5 \mathrm{cm} / \mathrm{s}^{2}\)
A marble is rolled so that it is projected horizontally off the top landing of a staircase. The initial speed of the marble is $3.0 \mathrm{m} / \mathrm{s} .\( Each step is \)0.18 \mathrm{m}\( high and \)0.30 \mathrm{m}$ wide. Which step does the marble strike first?
A runner times his speed around a track with a circumference of $0.50 \mathrm{mi} .\( He finds that he has run a distance of \)1.00 \mathrm{mi}$ in 4.0 min. What is his (a) average speed and (b) average velocity magnitude in \(\mathrm{m} / \mathrm{s}\) ?
Sheena can row a boat at \(3.00 \mathrm{mi} / \mathrm{h}\) in still water. She needs to cross a river that is 1.20 mi wide with a current flowing at $1.60 \mathrm{mi} / \mathrm{h} .$ Not having her calculator ready, she guesses that to go straight across, she should head \(60.0^{\circ}\) upstream. (a) What is her speed with respect to the starting point on the bank? (b) How long does it take her to cross the river? (c) How far upstream or downstream from her starting point will she reach the opposite bank? (d) In order to go straight across, what angle upstream should she have headed?
A tennis ball is thrown horizontally from an elevation of \(14.0 \mathrm{m}\) above the ground with a speed of \(20.0 \mathrm{m} / \mathrm{s}\) (a) Where is the ball after 1.60 s? (b) If the ball is still in the air, how long before it hits the ground and where will it be with respect to the starting point once it lands?
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