/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 75 Two soccer players, Mia and Alic... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 75

Two soccer players, Mia and Alice, are running as Alice passes the ball to Mia. Mia is running due north with a speed of 6.00 m/s. The velocity of the ball relative to Mia is 5.00 m/s in a direction 30.0\(^\circ\) east of south. What are the magnitude and direction of the velocity of the ball relative to the ground?

Short Answer

Expert verified
The ball's velocity relative to the ground is 3.01 m/s at 56.3掳 east of north.

Step by step solution

01

Understand the Problem

We need to find the velocity of the ball relative to the ground. We have two velocities: Mia's velocity (6.00 m/s north) and the ball's velocity relative to Mia (5.00 m/s 30.0掳 east of south). We want to determine the actual or absolute velocity of the ball as seen from the ground.
02

Resolve Ball's Relative Velocity into Components

The velocity of the ball relative to Mia needs to be split into its velocity components in the north-south and east-west directions. Since it's 30.0掳 east of south:\[ v_{b/M} = 5.00\, \text{m/s} \]- Southward component: \( v_{b/M, s} = 5.00 \times \cos(30.0掳) = 5.00 \times \frac{\sqrt{3}}{2} = 4.33 \text{ m/s south} \)- Eastward component: \( v_{b/M, e} = 5.00 \times \sin(30.0掳) = 5.00 \times 0.5 = 2.50 \text{ m/s east} \)
03

Combine Mia's Velocity with Ball's Components

Mia is moving north with a velocity of 6.00 m/s; therefore:- The northward component of the ball is Mia's northward velocity minus the southward component from the relative velocity:\[ v_{b, n} = 6.00 - 4.33 = 1.67 \text{ m/s north} \]- The eastward component from the ball relative to Mia remains as it is:\[ v_{b, e} = 2.50 \text{ m/s east} \]
04

Find the Resultant Velocity of the Ball

Now we have the components of the ball's velocity relative to the ground:- Northward component of 1.67 m/s- Eastward component of 2.50 m/sWe use the Pythagorean theorem to find the magnitude of the resultant velocity:\[ v_b = \sqrt{(1.67)^2 + (2.50)^2} \approx \sqrt{2.79 + 6.25} = \sqrt{9.04} \approx 3.01\, \text{m/s} \]
05

Calculate the Direction of the Ball's Velocity

To find the direction, we use the tangent function for the components:\[ \theta = \arctan\left(\frac{2.50}{1.67}\right) \approx \arctan(1.497) = 56.3掳 \]This angle is measured east of north (since the north component is the base and east is positive). Hence, the direction is 56.3掳 east of north.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Vector Components
When analyzing movement, it's crucial to break down velocities into vector components. Think of a vector as an arrow that shows both direction and magnitude (size). For Mia and the ball, we need to decompose the ball's velocity into two directions: north-south (vertical) and east-west (horizontal). In physics, vector components are like the legs of a right triangle, showing how a velocity splits into two perpendicular parts.
  • Example: If a ball's speed is 5 m/s, and it travels 30掳 east of south, the vector components tell us exactly how much of that speed is directed south and how much is east.
  • Think of using a compass to map out the direction: one hand points south, and the other east.
  • These directions help you understand the movement more precisely, converting a complex direction into simple vertical and horizontal speeds.
Understanding vector components lets you manage these directions mathematically by projecting them onto axes, making complex motion easier to handle.
Pythagorean Theorem
Once we have vector components, the Pythagorean Theorem is our go-to tool for recombining them to find the complete picture: the actual speed and path of the object. This theorem relates the sides of a right triangle, providing a straightforward method to calculate the hypotenuse (the longest side).
  • A right triangle has one 90-degree angle, which makes applying Pythagoras easy and powerful.
  • The formula is \( c = \sqrt{a^2 + b^2} \) where \( c \) is the hypotenuse, and \( a \) and \( b \) are the triangle's legs.
  • In our soccer exercise, the northward and eastward velocity components form a right triangle for the ball.
By using the Pythagorean Theorem, we find the magnitude (speed) of the ball's velocity relative to the ground as the hypotenuse, locking down its real movement path & speed.
Trigonometry in Physics
Trigonometry often steps in for final adjustments, especially for finding angles in physics problems. It uses ratios like sine, cosine, and tangent to relate the angles and sides of triangles, particularly right triangles. For these soccer players, we're interested in the tangent function to finalize the direction the ball moves relative to true north.
  • The tangent of an angle in a right triangle is the ratio of opposite side over adjacent side (\( \tan(\theta) = \frac{opposite}{adjacent} \)).
  • To pinpoint the ball鈥檚 direction, calculate the angle between the direction north and the actual path.
  • This process is critical when you need more than just speed鈥攗nderstanding direction gives a full picture of motion.
So when the math says the ball moves at 56.3掳 east of north, it's showing how trigonometry helps us project real-world movements onto a directional map, combining all components into a single, meaningful result.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An airplane is flying with a velocity of 90.0 m/s at an angle of 23.0\(^{\circ}\) above the horizontal. When the plane is 114 m directly above a dog that is standing on level ground, a suitcase drops out of the luggage compartment. How far from the dog will the suitcase land? Ignore air resistance.

At its Ames Research Center, NASA uses its large "20-G" centrifuge to test the effects of very large accelerations ("hypergravity") on test pilots and astronauts. In this device, an arm 8.84 m long rotates about one end in a horizontal plane, and an astronaut is strapped in at the other end. Suppose that he is aligned along the centrifuge's arm with his head at the outermost end. The maximum sustained acceleration to which humans are subjected in this device is typically 12.5\(g\). (a) How fast must the astronaut's head be moving to experience this maximum acceleration? (b) What is the \(difference\) between the acceleration of his head and feet if the astronaut is 2.00 m tall? (c) How fast in rpm (rev/min) is the arm turning to produce the maximum sustained acceleration?

An airplane is dropping bales of hay to cattle stranded in a blizzard on the Great Plains. The pilot releases the bales at 150 m above the level ground when the plane is flying at 75 m/s in a direction 55\(^{\circ}\) above the horizontal. How far in front of the cattle should the pilot release the hay so that the bales land at the point where the cattle are stranded?

A physics professor did daredevil stunts in his spare time. His last stunt was an attempt to jump across a river on a motorcycle \(\textbf{(Fig. P3.63). }\) The takeoff ramp was inclined at 53.0\(^{\circ}\), the river was 40.0 m wide, and the far bank was 15.0 m lower than the top of the ramp. The river itself was 100 m below the ramp. Ignore air resistance. (a) What should his speed have been at the top of the ramp to have just made it to the edge of the far bank? (b) If his speed was only half the value found in part (a), where did he land?

A model of a helicopter rotor has four blades, each 3.40 m long from the central shaft to the blade tip. The model is rotated in a wind tunnel at 550 rev/min. (a) What is the linear speed of the blade tip, in m/s? (b) What is the radial acceleration of the blade tip expressed as a multiple of \(g\)?
See all solutions

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Read Explanation

Measurements

Read Explanation

Rotational Dynamics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Torque and Rotational Motion

Read Explanation

Solid State Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.