/*! 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 29 A major-league pitcher can throw... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 29

A major-league pitcher can throw a baseball in excess of \(41.0 \mathrm{m} / \mathrm{s}\). If a ball is thrown horizontally at this speed, how much will it drop by the time it reaches a catcher who is \(17.0 \mathrm{m}\) away from the point of release?

Short Answer

Expert verified
The baseball will drop approximately 0.840 meters.

Step by step solution

01

Identify Known Values and Equation

We know the speed of the baseball is \(v = 41.0 \text{ m/s}\), and the distance to the catcher is \(d = 17.0 \text{ m}\). We need to find out how much the ball drops due to gravity before it reaches the catcher. We'll need to find the time of flight \(t\) first using the formula \(t = \frac{d}{v}\), because the horizontal velocity and distance is known.
02

Calculate Time of Flight

Substituting the known values in the formula \(t = \frac{d}{v}\), we have: \[ t = \frac{17.0}{41.0} = 0.4146 \text{ seconds.} \]
03

Use the Time of Flight to Determine Vertical Displacement

The vertical displacement (drop) \(y\) due to gravity after a time \(t\) can be calculated using the equation for motion under gravity: \[ y = \frac{1}{2} g t^2 \]where \(g = 9.81 \text{ m/s}^2\). Substituting the values, we get: \[ y = \frac{1}{2} \times 9.81 \times (0.4146)^2 \approx 0.840 \text{ meters.} \]
04

Conclusion

By following these calculations, the baseball will drop approximately \(0.840 \text{ meters}\) before it reaches the catcher.

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.

Horizontal Velocity
Horizontal velocity is all about movement along a straight, horizontal line. In our baseball scenario, the horizontal velocity is the speed at which the ball is thrown, which is given as 41.0 meters per second (m/s). Once an object is thrown horizontally, its speed in this direction remains constant unless some force acts on it, like air resistance.
  • Horizontal velocity does not change; gravity only affects vertical movement.
  • In most projectile motion problems, without the influence of air resistance, the object keeps its initial horizontal velocity throughout its flight.
Why is this important? Because knowing the horizontal velocity lets us calculate how long the ball will take to reach its target if we know the distance it has to cover.
Time of Flight
The time of flight is how long the baseball is in the air before it reaches the catcher. This is a vital piece of our puzzle because it impacts how far the ball falls vertically due to gravity.
To find the time of flight in this scenario, we used the formula:\[ t = \frac{d}{v} \]Here, `\(d\)` is the distance from the thrower to the catcher, **17.0 meters** in our problem, and `\(v\)` is the horizontal velocity, **41.0 m/s**. Substituting these values gives:\[ t \approx 0.4146 \text{ seconds} \]
  • This time tells us how long gravity has to affect the baseball.
  • Knowing this allows us to move on to calculating the vertical displacement.
Keeping track of time helps decode how the forces at play influence the baseball's motion.
Vertical Displacement
Vertical displacement is the distance an object moves vertically downward due to gravity while it is in motion. In the case of our baseball, it starts moving horizontally with no vertical speed, but gravity pulls it down.
Gravity's effect is calculated using the formula:\[ y = \frac{1}{2} g t^2 \]Here's what the symbols mean:
  • `\(y\)` is the vertical displacement.
  • `\(g\)` is the acceleration due to gravity, approximately 9.81 m/s虏 on Earth.
  • `\(t\)` is the time of flight you calculated earlier.
By substituting `\(t = 0.4146\)` seconds into the equation:\[ y \approx 0.840 \text{ meters} \]
This tells us the baseball drops approximately 0.84 meters by the time the catcher tries to catch it. Understanding vertical displacement is key to predicting the path of any projectile under the influence of gravity.

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

E A bird watcher meanders through the woods, walking \(0.50 \mathrm{km}\) due east, \(0.75 \mathrm{km}\) due south, and \(2.15 \mathrm{km}\) in a direction \(35.0^{\circ}\) north of west. The time required for this trip is \(2.50 \mathrm{h}\). Determine the magnitude and direction (relative to due west) of the bird watcher's (a) displacement and (b) average velocity. Use kilometers and hours for distance and time, respectively.

An airplane with a speed of \(97.5 \mathrm{m} / \mathrm{s}\) is climbing upward at an angle of \(50.0^{\circ}\) with respect to the horizontal. When the plane's altitude is \(732 \mathrm{m},\) the pilot releases a package. (a) Calculate the distance along the ground, measured from a point directly beneath the point of release, to where the package hits the earth. (b) Relative to the ground, determine the angle of the velocity vector of the package just before impact.

A rocket is fired at a speed of \(75.0 \mathrm{m} / \mathrm{s}\) from ground level, at an angle of \(60.0^{\circ}\) above the horizontal. The rocket is fired toward an \(11.0-\mathrm{m}\) -high wall, which is located \(27.0 \mathrm{m}\) away. The rocket attains its launch speed in a negligibly short period of time, after which its engines shut down and the rocket coasts. By how much does the rocket clear the top of the wall?

A golfer imparts a speed of \(30.3 \mathrm{m} / \mathrm{s}\) to a ball, and it travels the maximum possible distance before landing on the green. The tee and the green are at the same elevation. (a) How much time does the ball spend in the air? (b) What is the longest hole in one that the golfer can make, if the ball does not roll when it hits the green?

A radar antenna is tracking a satellite orbiting the earth. At a certain time, the radar screen shows the satellite to be \(162 \mathrm{km}\) away. The radar antenna is pointing upward at an angle of \(62.3^{\circ}\) from the ground. Find the \(x\) and \(y\) components (in \(\mathrm{km}\) ) of the position vector of the satellite, relative to the antenna.
See all solutions

Recommended explanations on Physics Textbooks

Medical Physics

Read Explanation

Torque and Rotational Motion

Read Explanation

Electricity

Read Explanation

Physics of Motion

Read Explanation

Turning Points in Physics

Read Explanation

Oscillations

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.