/*! 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 25 A steam catapult launches a jet ... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 25

A steam catapult launches a jet aircraft from the aircraft carrier John C. Stennis, giving it a speed of \(175 \mathrm{mi} / \mathrm{h}\) in \(2.50 \mathrm{~s}\). (a) Find the average acceleration of the plane. (b) Assuming the acceleration is constant, find the distance the plane moves.

Short Answer

Expert verified
The average acceleration of the plane is \(a\) m/s虏, and the plane moves \(d\) meters.

Step by step solution

01

Understand and Convert the Units

First, given that the speed is provided in miles per hour, it should be converted to meters per second, which is the SI unit of speed. To do this use the fact that 1 mi/h is roughly equal to 0.44704 m/s, so the final speed \(v_f = 175 * 0.44704\) m/s.
02

Calculate the Acceleration

To find the average acceleration, the equation \(a = (v_f - v_i) / t\) is used, where \(v_i\) is the initial speed, \(v_f\) is the final speed and \(t\) is the time. Since the plane is initially at rest, \(v_i = 0\). Thus, substitute \(0\) for \(v_i\), the converted speed for \(v_f\) and \(2.50\) s for \(t\) in the equation to get the acceleration.
03

Calculate the Distance

Assuming the acceleration is constant, the equation that connects distance, initial velocity, time, and acceleration is \(d = v_i*t + 0.5*a*t^2\). Since \(v_i = 0\), the equation is reduced to \(d = 0.5*a*t^2\). Substitute the value of \(a\) just calculated and \(t = 2.50\) s into this equation to obtain the distance that the plane travels.

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Key Concepts

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

Unit Conversion
When working with physics problems, it's key to have all quantities in consistent units. In this exercise, the speed of the aircraft is given in miles per hour (mi/h). However, in most physics equations, especially those involving acceleration and distance, you should use the metric system. That's where unit conversion comes into play.
  • One mile per hour can be converted to meters per second using the factor 0.44704.
  • For this problem, multiply the given speed of 175 mi/h by 0.44704 to get the speed in meters per second.
  • This conversion helps in applying the kinematic formulas correctly, which use meters and seconds as the base units.
This small change is crucial because units affect the outcome of every physics calculation. Ensuring that all measurements are in a consistent unit system can make or break your understanding and solution of a physics problem.
Kinematic Equations
Kinematic equations are a set of formulas that relate different aspects of motion such as velocity, acceleration, time, and distance. They are specifically used when the acceleration is constant, which fits the scenario in this exercise.
  • The basic kinematic equation used to find acceleration is: \( a = \frac{v_f - v_i}{t} \).
  • Here, \( v_f \) is the final velocity, \( v_i \) is the initial velocity, and \( t \) is the time taken.
  • For this aircraft launch, since it begins at rest, \( v_i = 0 \), simplifying the calculation. Thus, you only need to divide the final velocity by the time to find the average acceleration.
These kinematic equations are cornerstones of physics problems involving motion, especially in scenarios like this where we assume a simplistic model of constant acceleration.
Constant Acceleration
The concept of constant acceleration implies that the acceleration doesn't change over time. In our problem, this makes using kinematic equations straightforward. Once you know the acceleration, you can easily find other aspects of the object's motion.
  • Using constant acceleration, the distance covered can be calculated by the formula: \( d = v_i \cdot t + 0.5 \cdot a \cdot t^2 \).
  • Since the plane starts from rest, the initial velocity \( v_i \) is zero, simplifying this to: \( d = 0.5 \cdot a \cdot t^2 \).
  • Insert the previously found acceleration and time into this equation to calculate the distance.
Understanding constant acceleration not only helps in solving this specific problem, but also prepares you for tackling a wide range of physics problems that involve uniform motion.

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Most popular questions from this chapter

BIO Colonel John P. Stapp, USAF, participated in studying whether a jet pilot could survive emergency ejection. On March 19,1954 , he rode a rocketpropelled sled that moved down a track at a speed of \(632 \mathrm{mi} / \mathrm{h}\) (see Fig. P2.56). He and the sled were safely brought to rest in \(1.40 \mathrm{~s}\). Determine in SI units (a) the negative acceleration he experienced and (b) the distance he traveled during this negative acceleration.

An object moves with constant acceleration \(4.00 \mathrm{~m} / \mathrm{s}^{2}\) and over a time interval reaches a final velocity of \(12.0 \mathrm{~m} / \mathrm{s}\). (a) If its original velocity is \(6.00 \mathrm{~m} / \mathrm{s}\), what is its displacement during the time interval? (b) What is the distance it travels during this interval? (c) If its original velocity is \(-6.00 \mathrm{~m} / \mathrm{s}\), what is its displacement during this interval? (d) What is the total distance it travels during the interval in part (c)?

In 1865 Jules Verne proposed sending men to the Moon by firing a space capsule from a 220 -m-long cannon with final speed of \(10.97 \mathrm{~km} / \mathrm{s}\). What would have been the unrealistically large acceleration experienced by the space travelers during their launch? (A human can stand an acceleration of \(15 \mathrm{~g}\) for a short time.) Compare your answer with the free-fall acceleration, \(9.80 \mathrm{~m} / \mathrm{s}^{2}\).

An ice sled powered by a rocket engine starts from rest on a large frozen lake and accelerates at \(+40 \mathrm{ft} / \mathrm{s}^{2}\). After some time \(t_{1}\), the rocket engine is shut down and the sled moves with constant velocity \(v\) for a time \(t_{2}\). If the total distance traveled by the sled is \(17500 \mathrm{ft}\) and the total time is \(90 \mathrm{~s}\), find (a) the times \(t_{1}\) and \(t_{2}\) and (b) the velocity \(v\). At the \(17500-\mathrm{ft}\) mark, the sled begins to accelerate at \(-20 \mathrm{ft} / \mathrm{s}^{2}\). (c) What is the final position of the sled when it comes to rest? (d) How long does it take to come to rest?

One athlete in a race running on a long, straight track with a constant speed \(v_{1}\) is a distance \(d\) behind a second athlete running with a constant speed \(v_{2}\). (a) Under what circumstances is the first athlete able to overtake the second athlete? (b) Find the time \(t\) it takes the first athlete to overtake the second athlete, in terms of \(d, v_{1}\), and \(v_{2}\). (c) At what minimum distance \(d_{2}\) from the leading athlete must the finish line be located so that the trailing athlete can at least tie for first place? Express \(d_{2}\) in terms of \(d, v_{1}\), and \(v_{2}\) by using the result of part (b).
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