/*! 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 40 You throw a baseball straight up... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 40

You throw a baseball straight upward. The drag force is proportional to \(v^2\). In terms of \(g\), what is the y-component of the ball's acceleration when the ball's speed is half its terminal speed and (a) it is moving up? (b) It is moving back down?

Short Answer

Expert verified
(a) \( \frac{5g}{4} \) when moving up; (b) \( \frac{3g}{4} \) when moving down.

Step by step solution

01

Understand the Forces Involved

When a baseball is thrown upwards, two main forces act on it: gravitational force, which is constant and equals to the weight of the ball \( F_g = mg \), and the drag force, which depends on the velocity squared, given by \( F_d = kv^2 \), where \( k \) is a drag coefficient specific to the situation.
02

Understand Terminal Velocity

Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium prevents further acceleration. At terminal velocity, the drag force equals the gravitational force: \( kv_t^2 = mg \). It follows that \( k = \frac{mg}{v_t^2} \).
03

Half Terminal Velocity Scenario

If the speed of the ball is half of its terminal speed, then \( v = \frac{v_t}{2} \). Substituting this into the drag force equation gives: \[ F_d = k\left(\frac{v_t}{2}\right)^2 = \frac{mg}{v_t^2}\left(\frac{v_t^2}{4}\right) = \frac{mg}{4} \].
04

Calculate Net Force When Moving Up

When the ball is moving up, the gravitational force and the drag force act in the same direction (downwards). Thus, the net force is: \[ F_{net} = mg + \frac{mg}{4} = \frac{5mg}{4} \].
05

Calculate Acceleration When Moving Up

Using Newton's second law, \( F = ma \), the y-component of the ball's acceleration when moving up is:\[ a = \frac{F_{net}}{m} = \frac{5mg/4}{m} = \frac{5g}{4} \].
06

Calculate Net Force When Moving Down

When the ball is moving down, the gravitational force acts downwards, while the drag force acts upwards. Therefore, the net force is:\[ F_{net} = mg - \frac{mg}{4} = \frac{3mg}{4} \].
07

Calculate Acceleration When Moving Down

Using Newton's second law, the y-component of the ball's acceleration when moving down is:\[ a = \frac{F_{net}}{m} = \frac{3mg/4}{m} = \frac{3g}{4} \].

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.

Projectile Motion
Projectile motion occurs when an object is thrown into the air and moves under the influence of gravity. In this context, the motion is only in the vertical plane since the ball is thrown straight up. We must consider how the object moves up, reaches a peak height, and then moves back down. The constant force of gravity acts on the object throughout all of these stages, pulling it toward the earth. In this specific problem, analyzing the motion helps us understand the forces on the baseball at different points in its trajectory.
Drag Force
Drag force is a resistance force caused by the air in this scenario, acting opposite to the motion. It increases with the square of the velocity: the larger the speed, the greater the drag force. The formula for the drag force is given by:
  • \( F_d = kv^2 \)
Where \( k \) is a constant depending on the properties of the object and the air, and \( v \) is the velocity of the object. Drag force affects how quickly an object slows when it is moving up and also impacts the descent when moving down. Understanding drag is crucial for determining net forces in projectile motion, such as when calculating the baseball's changing acceleration.
Terminal Velocity
Terminal velocity is the maximum velocity an object can reach as it falls through a fluid, like air, when forces become balanced. As an object falls, it accelerates due to gravity. Eventually, the drag force due to air resistance increases to equal gravitational force. This results in no net force, so the object stops accelerating and continues to fall at constant speed, i.e., terminal velocity. For the baseball:
  • At terminal velocity, \( kv_t^2 = mg \).
This understanding helps in defining the context of the problem: calculating the acceleration when the velocity is not terminal but a fraction of it, such as half in our situation.
Newton's Second Law
Newton's Second Law states that the net force acting on an object is equal to the mass of that object multiplied by its acceleration: \( F = ma \). This principle is essential to solve any motion and force-related problems. In the context of this projectile motion, it helps to calculate the y-component of acceleration:
  • When moving up, both gravitational and drag forces act downwards.
  • When moving down, gravity pulls down and drag acts upwards.
Using this law, you calculate the acceleration both when the baseball is moving upwards and downwards by balancing these forces to find the net force and dividing by the mass of the object.

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

A bowling ball weighing 71.2 N 116.0 lb2 is attached to the ceiling by a 3.80-m rope. The ball is pulled to one side and released; it then swings back and forth as a pendulum. As the rope swings through the vertical, the speed of the bowling ball is 4.20 m/s. At this instant, what are (a) the acceleration of the bowling ball, in magnitude and direction, and (b) the tension in the rope?

While a person is walking, his arms swing through approximately a 45\(^\circ\) angle in \(\frac{1}{2}\) s. As a reasonable approximation, assume that the arm moves with constant speed during each swing. A typical arm is 70.0 cm long, measured from the shoulder joint. (a) What is the acceleration of a 1.0-g drop of blood in the fingertips at the bottom of the swing? (b) Draw a free-body diagram of the drop of blood in part (a). (c) Find the force that the blood vessel must exert on the drop of blood in part (a). Which way does this force point? (d) What force would the blood vessel exert if the arm were not swinging?

You tie a cord to a pail of water and swing the pail in a vertical circle of radius 0.600 m. What minimum speed must you give the pail at the highest point of the circle to avoid spilling water?

(a) If the coefficient of kinetic friction between tires and dry pavement is 0.80, what is the shortest distance in which you can stop a car by locking the brakes when the car is traveling at 28.7 m/s (about 65 mi/h)? (b) On wet pavement the coefficient of kinetic friction may be only 0.25. How fast should you drive on wet pavement to be able to stop in the same distance as in part (a)? (\(Note\): Locking the brakes is \(not\) the safest way to stop.)

A flat (unbanked) curve on a highway has a radius of 170.0 m. A car rounds the curve at a speed of 25.0 m/s. (a) What is the minimum coefficient of static friction that will prevent sliding? (b) Suppose that the highway is icy and the coefficient of static friction between the tires and pavement is only one- third of what you found in part (a). What should be the maximum speed of the car so that it can round the curve safely?
See all solutions

Recommended explanations on Physics Textbooks

Circular Motion and Gravitation

Read Explanation

Kinematics Physics

Read Explanation

Fundamentals of Physics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Force

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.