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Chapter 4: Problem 72

A rugby player runs with the ball directly toward his opponent's goal, along the positive direction of an \(x\) axis. He can legally pass the ball to a teammate as long as the ball's velocity relative to the field does not have a positive \(x\) component. Suppose the player runs at speed \(4.0 \mathrm{~m} / \mathrm{s}\) relative to the field while he passes the ball with velocity \(\vec{v}_{B P}\) relative to himself. If \(\vec{v}_{B P}\) has magnitude \(6.0 \mathrm{~m} / \mathrm{s},\) what is the smallest angle it can have for the pass to be legal?

Short Answer

Expert verified
The smallest angle \(\theta\) for a legal pass is approximately \(131.8^\circ\).

Step by step solution

01

Identify Given Data

The speed of the rugby player relative to the field is given as \(4.0 \text{ m/s}\). The velocity of the ball relative to the player, \(\vec{v}_{BP}\), has a magnitude of \(6.0 \text{ m/s}\). To make a legal pass, the ball's velocity relative to the field should not have a positive \(x\) component. We need to find the smallest angle \(\theta\) such that the \(x\) component of the ball's velocity relative to the field is zero or negative.
02

Understand Velocity Relationship

The velocity of the ball relative to the field \(\vec{v}_{BF}\) can be expressed as the sum of the player's velocity relative to the field and the ball's velocity relative to the player: \(\vec{v}_{BF} = \vec{v}_{PF} + \vec{v}_{BP}\). Here, \(\vec{v}_{PF}\) represents the player's velocity, which is \(4.0 \text{ m/s}\) along the \(x\) axis.
03

Express Velocity Components

The velocity of the ball relative to the player can be broken into components: the \(x\) component is \(6.0 \cos \theta\) and the \(y\) component is \(6.0 \sin \theta\). Thus, the \(x\) component of the ball's velocity relative to the field is given by \(4.0 + 6.0 \cos \theta\).
04

Set Up Equations for Legal Pass

For a legal pass, the \(x\) component of the ball鈥檚 velocity relative to the field must be non-positive: \[4.0 + 6.0 \cos \theta \leq 0\]
05

Solve for \(\theta\)

Rearrange the inequality to find \(\cos \theta\): \[6.0 \cos \theta \leq -4.0\]This reduces to: \[\cos \theta \leq -\frac{2}{3}\]Find the smallest angle \(\theta\) by determining the inverse cosine of \(-\frac{2}{3}\). \(\theta \approx \cos^{-1}(-\frac{2}{3}) \approx 131.8^\circ\).

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Key Concepts

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

Relative Velocity
In physics, relative velocity is an essential concept that simplifies the analysis of moving objects in relation to each other. It considers the velocity of an object as observed from a different moving reference frame.
The concept becomes particularly useful when solving problems where objects are in motion relative to different observers. In the context of the rugby problem, the ball's velocity is considered relative to the player and the field.
  • The rugby player runs at a constant speed along the field.
  • The ball's velocity, as perceived by the player, is different from how it's perceived on the field.
  • Adding the player's velocity to the ball's relative velocity results in the ball's velocity relative to the field.
Understanding how velocity transforms between reference frames helps to determine if the ball satisfies conditions for a legal pass.
Vector Components
Breaking vectors into components is a fundamental part of physics problem-solving. It allows for easier manipulation and calculation, especially when vectors are at an angle. A vector can be represented through its components along the axes, such as the horizontal (x) and vertical (y) components.The velocity vector of the ball, thrown at an angle, is divided into:
  • Horizontal component: This is given by \(6.0 \cos \theta\).
  • Vertical component: Represented by \(6.0 \sin \theta\).
The horizontal component is crucial here because it impacts whether the ball travels with or against the player's motion. The legal pass condition ensures the x-component of the ball's velocity does not exceed a certain threshold.
Physics Problem Solving
Physics problem-solving often involves identifying known and unknown quantities, setting up relationships, and using principles to navigate through formulas. The given rugby problem demonstrates this through systematic steps.The importance of carefully identifying quantities:
  • Recognizing known parameters: player's and ball's velocity.
  • Determining what is to be found: the minimum angle \(\theta\).
Setting up the Equation:
  • Form the equation by adding vector components of velocities.
  • Ensure conditions for legality by using inequalities.
This process highlights the importance of breaking problems into manageable parts and dealing with each aspect systematically.
Angular Calculation
Angles play a crucial role in the analysis of motion, especially when vectors are concerned. Calculating the angle in scenarios involving motion can determine whether conditions, like legality, are met.To find the minimum angle \(\theta\) in the problem:
  • Start by arranging the inequality: \(4.0 + 6.0 \cos \theta \leq 0\).
  • Rearrange to find the expression for \(\cos \theta\): \(\cos \theta \leq -\frac{2}{3}\).
  • Solving for \(\theta\) requires using inverse trigonometric functions: \(\theta \approx \cos^{-1}(-\frac{2}{3})\).
This step ensures the player's pass meets the criteria through precise angular calculation.

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Most popular questions from this chapter

An astronaut is rotated in a horizontal centrifuge at a radius of \(5.0 \mathrm{~m} .\) (a) What is the astronaut's speed if the centripetal acceleration has a magnitude of \(7.0 g ?\) (b) How many revolutions per minute are required to produce this acceleration? (c) What is the period of the motion?

A frightened rabbit moving at \(6.00 \mathrm{~m} / \mathrm{s}\) due east runs onto a large area of level ice of negligible friction. As the rabbit slides across the ice, the force of the wind causes it to have a constant acceleration of \(1.40 \mathrm{~m} / \mathrm{s}^{2},\) due north. Choose a coordinate system with the origin at the rabbit's initial position on the ice and the positive \(x\) axis directed toward the east. In unit-vector notation, what are the rabbit's (a) velocity and (b) position when it has slid for 3.00 s?

You are to throw a ball with a speed of \(12.0 \mathrm{~m} / \mathrm{s}\) at a target that is height \(h=5.00 \mathrm{~m}\) above the level at which you release the ball (Fig. \(4-58\) ). You want the ball's velocity to be horizontal at the instant it reaches the target. (a) At what angle \(\theta\) above the horizontal must you throw the ball? (b) What is the horizontal distance from the release point to the target? (c) What is the speed of the ball just as it reaches the target?

A boat is traveling upstream in the positive direction of an \(x\) axis at \(14 \mathrm{~km} / \mathrm{h}\) with respect to the water of a river. The water is flowing at \(9.0 \mathrm{~km} / \mathrm{h}\) with respect to the ground. What are the (a) magnitude and (b) direction of the boat's velocity with respect to the ground? A child on the boat walks from front to rear at \(6.0 \mathrm{~km} / \mathrm{h}\) with respect to the boat. What are the (c) magnitude and (d) direction of the child's velocity with respect to the ground?

For women's volleyball the top of the net is \(2.24 \mathrm{~m}\) above the floor and the court measures \(9.0 \mathrm{~m}\) by \(9.0 \mathrm{~m}\) on each side of the net. Using a jump serve, a player strikes the ball at a point that is \(3.0 \mathrm{~m}\) above the floor and a horizontal distance of \(8.0 \mathrm{~m}\) from the net. If the initial velocity of the ball is horizontal, (a) what minimum magnitude must it have if the ball is to clear the net and (b) what maximum magnitude can it have if the ball is to strike the floor inside the back line on the other side of the net?
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