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Chapter 3: Problem 28

Baseball player A bunts the ball by hitting it in such a way that it acquires an initial velocity of \(1.9 \mathrm{~m} / \mathrm{s}\) parallel to the ground. Upon contact with the bat the ball is \(1.2 \mathrm{~m}\) above the ground. Player B wishes to duplicate this bunt, in so far as he also wants to give the ball a velocity parallel to the ground and have his ball travel the same horizontal distance as player A's ball does. However, player \(\mathrm{B}\) hits the ball when it is \(1.5 \mathrm{~m}\) above the ground. What is the magnitude of the initial velocity that player B's ball must be given?

Short Answer

Expert verified
Player B must give the ball an initial velocity of approximately 1.70 m/s.

Step by step solution

01

Determine the time ball A is in the air

Player A's ball was hit from a height of 1.2 meters. To find the time (t) it stays in the air before hitting the ground, we use the equation of motion for free fall: \[ h = \frac{1}{2}gt^2 \]where \(h = 1.2\,\mathrm{m} \) and \(g = 9.8\,\mathrm{m/s^2} \). Solving for \(t\), we get:\[ 1.2 = \frac{1}{2} \times 9.8 \times t^2 \]\[ t^2 = \frac{2 \times 1.2}{9.8} \]\[ t = \sqrt{\frac{2.4}{9.8}} \approx 0.495 \text{ seconds} \].
02

Calculate the horizontal distance traveled by ball A

The horizontal distance (d) traveled by ball A can be found by multiplying its velocity by the time in the air:\[ d = v \times t = 1.9\,\mathrm{m/s} \times 0.495\,\mathrm{s} \]\[ d \approx 0.941\,\mathrm{m} \].
03

Determine the time ball B is in the air

Player B's ball is hit from a height of 1.5 meters. Using the same equation for free fall, we determine the time \(t'\) it stays in the air:\[ 1.5 = \frac{1}{2} \times 9.8 \times (t')^2 \]\[ (t')^2 = \frac{3.0}{9.8} \]\[ t' = \sqrt{\frac{3.0}{9.8}} \approx 0.553 \text{ seconds} \].
04

Calculate the necessary initial velocity for ball B

To travel the same horizontal distance as ball A (\(0.941\,\mathrm{m}\)), ball B's initial velocity \(v'\) must satisfy:\[ d = v' \times t' \]\[ 0.941 = v' \times 0.553 \]Solving for \(v'\), we have:\[ v' = \frac{0.941}{0.553} \approx 1.70\,\mathrm{m/s} \].

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Key Concepts

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

Bunt Mechanics
Bunt mechanics in baseball involve a precise contact between the bat and the ball. The aim is to change the direction of the ball subtly so that it travels only slightly forward. This requires skill in positioning the bat correctly, as well as timing the impact to minimize power. A successful bunt positions the ball to fall close to the ground, making it challenging for fielders to react quickly.
  • The bat should meet the ball with minimal motion to avoid excessive speed.
  • Positioning is key, adjusting the bat angle to control the ball's direction.
  • Complete mastery of bunt mechanics results in controlled velocity, allowing the ball to travel over a set horizontal distance before touching down.
Understanding the mechanics behind bunting facilitates effective execution, which heavily relies on hand-eye coordination and minimal force application. In the scenario of Player A and Player B, both utilize bunt mechanics to initiate their ball's motion at varying initial velocities.
Free Fall Equations
Free fall equations help calculate the time it takes for an object to hit the ground, assuming it has only the force of gravity acting upon it. In the context of baseball, the free fall equations are useful to determine how long the ball, after being bunted, stays in the air before it lands. The physics behind free fall can be simplified by the equation: \[ h = \frac{1}{2}gt^2 \]- **Where:** - \( h \) is the height from which the ball is dropped. - \( g \) is the acceleration due to gravity, approximately \( 9.8\,\mathrm{m/s^2} \). - \( t \) is the time in seconds.Upon solving this equation for time, students can comprehend how different bunt heights (1.2 meters vs. 1.5 meters) affect the time a ball stays airborne. This comprehension helps not just in theoretical exercises but also in practical field applications where precise timing dictates the outcome of the play. Applying the formula provides a fundamental understanding of gravity's impact on projectile motion.
Horizontal Distance Calculation
Calculating horizontal distance in projectile motion involves understanding both initial velocity and the time the object remains in motion. The formula to calculate the horizontal distance \( d \) is straightforward: \[ d = v \, t \]- **Where:** - \( d \) refers to the total horizontal distance covered. - \( v \) equates to the horizontal component of initial velocity. - \( t \) is the time duration the object is airborne.In baseball bunting, this calculation determines how far the bunted ball will land. From Player A's scenario, one can see that even a slight change in vertical drop height requires the initial velocity to be carefully adjusted, as seen when Player B effectively replicated the same horizontal distance. This core concept reinforces the crucial relation between travel time and velocity, emphasizing the significance of precision in both sports and physics. Calculations ensure skillful bunt execution while illustrating the close interplay between airborne time and velocity in achieving desired distances.

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Most popular questions from this chapter

(a) A projectile is launched at a speed \(v_{0}\) and at an angle above the horizontal; its initial velocity components are \(v_{0_{x}}\) and \(v_{0_{y}}\). In the absence of air resistance, what is the speed of the projectile at the peak of its trajectory? (b) What is its speed just before it lands at the same vertical level from which it was launched? (c) Consider its speed at a point that is at a vertical level between that in (a) and (b). How does the speed at this point compare with the speeds identified in (a) and (b)? In each case, give your reasoning. A golfer hits a shot to a green that is elevated \(3.0 \mathrm{~m}\) above the point where the ball is struck. The ball leaves the club at a speed of \(14.0 \mathrm{~m} / \mathrm{s}\) at an angle of \(40.0^{\circ}\) above the horizontal. It rises to its maximum height and then falls down to the green. Ignoring air resistance, find the speed of the ball just before it lands. Check to see that your answer is consistent with your answer to part (c) of the Concept Questions.

Useful background for this problem can be found in Multiple-Concept Example \(2 .\) On a spacecraft two engines fire for a time of \(565 \mathrm{~s}\). One gives the craft an acceleration in the \(x\) direction of \(a_{x}=5.10 \mathrm{~m} / \mathrm{s}^{2},\) while the other produces an acceleration in the \(y\) direction of \(a_{y}=7.30 \mathrm{~m} / \mathrm{s}^{2} .\) At the end of the firing period, the craft has velocity components of \(v_{x}=3775 \mathrm{~m} / \mathrm{s}\) and \(v_{y}=4816 \mathrm{~m} / \mathrm{s} .\) Find the magnitude and direction of the initial velocity. Express the direction as an angle with respect to the \(+x\) axis.

Interactive Solution \(\underline{3.11}\) at presents a model for solving this problem. The earth moves around the sun in a nearly circular orbit of radius \(1.50 \times 10^{11} \mathrm{~m} .\) During the three summer months (an elapsed time of \(7.89 \times 10^{6} \mathrm{~s}\) ), the earth moves one-fourth of the distance around the sun. (a) What is the average speed of the earth? (b) What is the magnitude of the average velocity of the earth during this period?

Michael Jordan, formerly of the Chicago Bulls basketball team, had some fanatic fans. They claimed that he was able to jump and remain in the air for two full seconds from launch to landing. Evaluate this claim by calculating the maximum height that such a jump would attain. For comparison, Jordan's maximum jump height has been estimated at about one meter.

Two passenger trains are passing each other on adjacent tracks. Train A is moving east with a speed of \(13 \mathrm{~m} / \mathrm{s}\), and train \(\mathrm{B}\) is traveling west with a speed of \(28 \mathrm{~m} / \mathrm{s}\). (a) What is the velocity (magnitude and direction) of train A as seen by the passengers in train B? (b) What is the velocity (magnitude and direction) of train B as seen by the passengers in train A?
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