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Chapter 2: Problem 21

The fastest measured pitched baseball left the pitcher's hand at a speed of \(45.0 \mathrm{~m} / \mathrm{s}\). If the pitcher was in contact with the ball over a distance of \(1.50 \mathrm{~m}\) and produced constant acceleration, (a) what acceleration did he give the ball, and (b) how much time did it take him to pitch it?

Short Answer

Expert verified
The acceleration the pitcher gives to the ball is \(675 \mathrm{~m/s}^2\) and the time it took him to pitch it is \(0.067 \mathrm{s}\).

Step by step solution

01

Derive the acceleration formula

To figure out the acceleration, we need to use the equation \(v^2= u^2 + 2as\). Because the ball starts from rest in the pitcher's hand, the initial velocity (\(u\)) is 0. Substituting the given values \(v = 45.0 \mathrm{~m} / \mathrm{s}\) and \(s = 1.50 \mathrm{~m}\), we get \(45.0^2 = 0^2 + 2 * a * 1.50\). Simplifying the equation, we get \(2 * a * 1.50 = 45.0^2\). Solving for \(a\) we get \(a = 45.0^2 / (2 * 1.50)\) which calculates to \(a = 675 \mathrm{~m} / \mathrm{s}^2\).
02

Compute the time taken

Now that we have the acceleration, we can use the equation \(v = u + at\) to determine the time it took to pitch the ball. Again, the initial velocity (\(u\)) is 0. Substituting the known values \(v = 45.0 \mathrm{~m} / \mathrm{s}\) and \(a = 675 \mathrm{~m} / \mathrm{s}^2\), we get \(45.0 = 0 + 675 * t\). Solving for \(t\), we get \(t = 45.0 / 675\) which calculates to \(t = 0.067 \mathrm{s}\).

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Key Concepts

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

Constant Acceleration
When we discuss constant acceleration in physics, especially in the context of baseball pitching, we refer to the steady increase in velocity of an object in a linear fashion over time. This situation implies that the speeding-up process occurs at an unchanging rate. For example, if a baseball picks up speed uniformly as it leaves the pitcher's hand, it is undergoing constant acceleration.

Applying constant acceleration to the problem of pitching, we can use the pitcher's arm and the ball as our system. The distance over which the ball accelerates is limited to the pitcher's release, which in the provided exercise is given as 1.50 meters. If the ball's velocity increases from rest to 45.0 m/s within this distance, then the acceleration can be determined using appropriate kinematic equations - ensuring that the acceleration is constant, simplifies the scenario allowing us to predict the ball鈥檚 behavior with a simple mathematical model.
Kinematic Equations
The kinematic equations are a set of four equations that relate the five kinematic variables: acceleration (a), time (t), displacement (s), initial velocity (u), and final velocity (v). These equations are essential tools for solving problems involving motion in one dimension with constant acceleration, such as a pitcher throwing a baseball.

For instance, the equation used in the exercise provided, \(v^2= u^2 + 2as\), is one of these kinematic equations. It is particularly useful when time is not a factor in our calculation. In the example, it allows us to solve for acceleration directly given the final velocity and the displacement. It's crucial, however, to apply the correct equation based on the given and sought quantities, and to remember that these formulas are only valid when acceleration is constant throughout the motion.
Motion in One Dimension
Motion that follows a straight line path is known as motion in one dimension. This type of motion is the simplest form to analyze and has wide applications including the study of objects moving vertically under the influence of gravity or vehicles moving along a straight road.

In the context of baseball pitching, we consider the motion in one dimension horizontally, from the pitcher's hand towards the catcher's mitt. Assuming no other forces acting on the ball, like air resistance or spin, the ball will accelerate in a straight line. The concept simplifies the analysis of the problem by focusing on the linear path of the ball and using kinematic equations to describe and predict the ball's behavior, as highlighted in the exercise.

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Most popular questions from this chapter

In the first stage of a two-stage rocket, the rocket is fired from the launch pad starting from rest but with a constant acceleration of \(3.50 \mathrm{~m} / \mathrm{s}^{2}\) upward. At \(25.0 \mathrm{~s}\) after launch, the second stage fires for \(10.0 \mathrm{~s}\), which boosts the rocket's velocity to \(132.5 \mathrm{~m} / \mathrm{s}\) upward at \(35.0 \mathrm{~s}\) after launch. This firing uses up all of the fuel, however, so after the second stage has finished firing, the only force acting on the rocket is gravity. Ignore air resistance. (a) Find the maximum height that the stage-two rocket reaches above the launch pad. (b) How much time after the end of the stage-two firing will it take for the rocket to fall back to the launch pad? (c) How fast will the stagetwo rocket be moving just as it reaches the launch pad?

A Honda Civic travels in a straight line along a road. The car's distance \(x\) from a stop sign is given as a function of time \(t\) by the equation \(x(t)=\alpha t^{2}-\beta t^{3},\) where \(\alpha=1.50 \mathrm{~m} / \mathrm{s}^{2}\) and \(\beta=0.0500 \mathrm{~m} / \mathrm{s}^{3} .\) Calculate the average velocity of the car for each time interval: (a) \(t=0\) to \(t=2.00 \mathrm{~s} ;\) (b) \(t=0\) to \(t=4.00 \mathrm{~s} ;\) (c) \(t=2.00 \mathrm{~s}\) to \(t=4.00 \mathrm{~s}\)

A hot-air balloonist, rising vertically with a constant velocity of magnitude \(5.00 \mathrm{~m} / \mathrm{s},\) releases a sandbag at an instant when the balloon is 40.0 \(\mathrm{m}\) above the ground (Fig. \(\mathbf{E} 2.42\) ). After the sandbag is released, it is in free fall. (a) Compute the position and velocity of the sandbag at \(0.250 \mathrm{~s}\) and \(1.00 \mathrm{~s}\) after its release. (b) How many seconds after its release does the bag strike the ground? (c) With what magnitude of velocity does it strike the ground? (d) What is the greatest height above the ground that the sandbag reaches? (e) Sketch \(a_{y}-t, v_{y}-t,\) and \(y-t\) graphs for the motion.

An astronaut has left the International Space Station to test a new space scooter. Her partner measures the following velocity changes, each taking place in a \(10 \mathrm{~s}\) interval. What are the magnitude, the algebraic sign, and the direction of the average acceleration in each interval? Assume that the positive direction is to the right. (a) At the beginning of the interval, the astronaut is moving toward the right along the \(x\) -axis at \(15.0 \mathrm{~m} / \mathrm{s}\), and at the end of the interval she is moving toward the right at \(5.0 \mathrm{~m} / \mathrm{s}\). (b) At the beginning she is moving toward the left at \(5.0 \mathrm{~m} / \mathrm{s},\) and at the end she is moving toward the left at \(15.0 \mathrm{~m} / \mathrm{s}\). (c) At the beginning she is moving toward the right at \(15.0 \mathrm{~m} / \mathrm{s}\), and at the end she is moving toward the left at \(15.0 \mathrm{~m} / \mathrm{s}\).

A juggler throws a bowling pin straight up with an initial speed of \(8.20 \mathrm{~m} / \mathrm{s}\). How much time elapses until the bowling pin returns to the juggler's hand?
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