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Chapter 4: Problem 47

A golfer takes three putts to get the ball into the hole. The first putt displaces the ball \(3.66 \mathrm{~m}\) north, the second \(1.83 \mathrm{~m}\) southeast, and the third \(0.91 \mathrm{~m}\) southwest. What are (a) the magnitude and (b) the direction of the displacement needed to get the ball into the hole on the first putt?

Short Answer

Expert verified
The magnitude is 4.73 m, and the direction is 113.9掳.

Step by step solution

01

- Represent the displacement vectors

Represent the displacements as vectors. The first putt is 3.66 m north, which can be written as \(\textbf{A} = 3.66 \hat{j}\). The second putt is 1.83 m southeast which makes an angle of 135掳 with the positive x-axis, so \(\textbf{B} = 1.83 \cos(135掳) \hat{i} + 1.83 \sin(135掳) \hat{j}\). The third putt is 0.91 m southwest which makes an angle of 225掳 with the positive x-axis, thus \(\textbf{C} = 0.91 \cos(225掳) \hat{i} + 0.91 \sin(225掳) \hat{j}\).
02

- Calculate the components of each vector

Determine the x and y components of each vector:\(\textbf{A}\) has components: \(\textbf{A}_x = 0\), \(\textbf{A}_y = 3.66\footnotesize m\) \(\textbf{B}\) has components: \(\textbf{B}_x = 1.83 \cos(135掳)\footnotesize m = -1.29\footnotesize m\), \(\textbf{B}_y = 1.83 \sin(135掳)\footnotesize m = 1.29\footnotesize m\) \(\textbf{C}\) has components: \(\textbf{C}_x = 0.91 \cos(225掳)\footnotesize m = -0.643\footnotesize m\), \(\textbf{C}_y = 0.91 \sin(225掳)\footnotesize m = -0.643\footnotesize m\)
03

- Find the resultant vector sum

Add the components of the three vectors to find the resultant vector:\(\textbf{R}_x = \textbf{A}_x + \textbf{B}_x + \textbf{C}_x = 0 - 1.29 - 0.643 = -1.933\footnotesize m\)\(\textbf{R}_y = \textbf{A}_y + \textbf{B}_y + \textbf{C}_y = 3.66 + 1.29 - 0.643 = 4.307\footnotesize m\)
04

- Calculate magnitude of the resultant displacement

Use the Pythagorean theorem to find the magnitude of the displacement:\[ | \textbf{R} | = \sqrt{(\textbf{R}_x)^2 + (\textbf{R}_y)^2} = \sqrt{(-1.933)^2 + (4.307)^2} = 4.73\ \text{m} \]
05

- Determine the direction of the resultant displacement

Find the direction by calculating the angle with the positive x-axis:\[ \theta = tan^{-1} (\frac{\textbf{R}_y}{\textbf{R}_x}) = tan^{-1}(\frac{4.307}{-1.933}) = tan^{-1}(-2.22) = 113.9掳 \] The direction is 113.9掳 from the positive x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

displacement vector
In physics, a displacement vector represents the change in position of an object. It has both magnitude and direction. For instance, in the exercise, the golfer's first putt moves the ball 3.66 meters north. This can be represented as a vector pointing directly upwards (along the y-axis) with a length of 3.66 meters. The displacement vector shows not only how far the ball moves but also in which direction. This makes it different from just 'distance', which does not include direction. This precision is vital in physics to analyze movements and find out the exact position changes.
resultant vector
When multiple vectors are involved, the resultant vector is their combined effect. In the exercise, the golfer's ball is moved by three separate displacements. To find out the single displacement that would have the same effect as these three combined, we need to calculate the resultant vector. This involves adding the individual vectors together. The resultant vector gives us the total displacement of the ball considering the combined effect of all three putts. It shows the final position change from the initial point to the hole.
Pythagorean theorem
The Pythagorean theorem is a fundamental principle in mathematics often used in physics to calculate the magnitude of vectors. It is expressed as \[c = \sqrt{a^2 + b^2} \]. In the context of the exercise, after finding the components of the resultant vector, we apply the Pythagorean theorem. By plugging in the resultant x and y components, \(-1.933\footnotesize m\) and \(4.307\footnotesize m\), we calculate the magnitude of the total displacement: \(\textbf{R} = \sqrt{(-1.933)^2 + (4.307)^2} \). This gives us approximately 4.73 meters, which is the magnitude of the resultant displacement vector.
trigonometric functions
Trigonometric functions, like sine and cosine, are crucial to resolving vectors into their components. For instance, in the exercise, we need these functions to break down the second and third putts into x and y components because they are not aligned with the axes. For example, the second putt at 135掳 is resolved using \(1.83 \cos(135掳) \hat{i} + 1.83 \sin(135掳) \hat{j}\), resulting in x and y components: \(-1.29\footnotesize m\) and \(1.29\footnotesize m\). These components are then used to compute the resultant displacement. Understanding trigonometric functions is vital for working with vectors in physics.
direction of a vector
The direction of a vector is essential as it tells us where the vector points. In the exercise, we use the arctangent function to find the direction of the resultant displacement. The formula used here is \[ \theta = \tan^{-1} (\frac{R_y}{R_x}) \], which calculates the angle \(\theta\) relative to the positive x-axis. With \( R_x = -1.933\footnotesize m \) and \( R_y = 4.307\footnotesize m \), we get \(\theta \approx \ 113.9掳\). This angle reveals the exact direction of the resultant vector, ensuring our analysis captures both how far and in which direction the ball has been displaced.

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Most popular questions from this chapter

Prove that two vectors must have equal magnitudes if their sum is perpendicular to their difference.

What are (a) the \(x\) -component and (b) the \(y\) -component of a vector \(\vec{a}\) in the \(x y\) plane if its direction is \(250^{\circ}\) counterclockwise from the positive direction of the \(x\) axis and its magnitude is \(7.3 \mathrm{~m}\) ?

A person desires to reach a point that is \(3.40 \mathrm{~km}\) from her present location and in a direction that is \(35.0^{\circ}\) north of east. However, she must travel along streets that are oriented either north-south or east-west. What is the minimum distance she could travel to reach her destination?

Find the sum of the following four vectors in (a) unit-vector notation, and as (b) a magnitude and (c) an angle relative to \(+x\). \(\vec{P}: 10.0 \mathrm{~m}\), at \(25.0^{\circ}\) counterclockwise from \(+x\) \(\vec{Q}: 12.0 \mathrm{~m}\), at \(10.0^{\circ}\) counterclockwise from \(+y\) \(\vec{R}: 8.00 \mathrm{~m}\), at \(20.0^{\circ}\) clockwise from \(-y\) \(\vec{S}: 9.00 \mathrm{~m}\), at \(40.0^{\circ}\) counterclockwise from \(-y\)

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