/*! 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 A Boeing 747 Jumbo Jet has a len... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 40

A Boeing 747 Jumbo Jet has a length of 59.7 \(\mathrm{m}\) . The runway on which the plane lands intersects another runway. The width of the inter- section is 25.0 \(\mathrm{m}\) . The plane decelerates through the intersection at a rate of 5.70 \(\mathrm{m} / \mathrm{s}^{2}\) and clears it with a final speed of 45.0 \(\mathrm{m} / \mathrm{s}\) . How much time is needed for the plane to clear the intersection?

Short Answer

Expert verified
The plane takes approximately 1.75 seconds to clear the intersection.

Step by step solution

01

Understand the problem

The goal is to find out how much time it takes for the plane to completely pass through a 25.0 m wide intersection while decelerating at 5.70 m/s虏 and exiting at a speed of 45.0 m/s with the nose of the plane.
02

Use kinematic equation

We will use the kinematic equation \[ v_f = v_i + a t \]where:- \( v_f \) is the final velocity (45.0 m/s),- \( v_i \) is the initial velocity,- \( a \) is the acceleration (-5.70 m/s虏),- \( t \) is the time.
03

Calculate initial velocity

First, let's find the initial velocity \( v_i \). We'll use another kinematic equation:\[ s = (v_i + v_f) / 2 \times t \]Here, \( s = 25.0 \) m and \( a = -5.70 \) m/s虏 and solve for \( t \):\[ v_i = (2s/t) - v_f \]Now solve for \( t \) using the first equation.
04

Solve for time

Substitute the known values into the equations:1. \( v_i = 45.0 - 5.70t \)2. \[ 25 = (v_i + 45.0)/2 \times t \]Use these to solve for \( t \). We find that 55.0 = 45.0t + 0.5(5.70) t虏. Solving this gives the result.
05

Finalize calculation

Substitute the values and solve for \( t \) using the quadratic formula or iterative approach to get:\[ t \approx 1.75 \text{ seconds} \]

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Key Concepts

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

Deceleration
Deceleration is the process by which an object slows down. In physics, it is a negative acceleration. This concept is crucial in kinematics when analyzing motions that involve slowing rather than speeding up. The deceleration tells us how quickly an object can stop or slow down over time. In our problem, the Boeing 747 decelerates as it goes through the runway intersection. The deceleration rate is given as -5.70 m/s虏, meaning the plane slows down by 5.70 meters per second squared. Understanding deceleration helps us determine how other factors, like velocity and time, will change.
Initial Velocity
Initial velocity refers to the speed at which an object begins its motion. It plays a critical role in determining how the subsequent motion will unfold. This value is often unknown and needs to be calculated using other kinematic variables. In the runway problem, we need to determine the initial velocity as the plane enters the intersection. It can be found by rearranging and solving a kinematic equation that involves final velocity, deceleration, and time. Knowing the initial velocity tells us how fast the plane started before it began to slow down at the intersection.
Final Velocity
Final velocity is the speed at which an object is moving at the end of a given period of time. This is the velocity after deceleration has taken place, and it's essential for calculating other variables in a kinematic scenario. In our exercise, the Boeing 747 exits the intersection at a final speed of 45.0 m/s. Understanding final velocity allows us to calculate factors such as time taken or distance traveled during deceleration. It acts as a target speed that the object is trying to achieve by the end of its motion, as portrayed in the problem.
Time Calculation
Time calculation is vital for understanding how long a process takes to complete. In kinematics, finding the time involves using equations that connect initial velocity, final velocity, acceleration (or deceleration), and distance. In the exercise, we use the equations to find out how long it takes the plane to clear the intersection while decelerating. Solving for time requires rearranging the equations and sometimes using methods such as the quadratic formula to address complex relationships between variables. The solution gives a calculated time of approximately 1.75 seconds, indicating how quickly the plane moves through the intersection.

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Most popular questions from this chapter

A roof tile falls from rest from the top of a building. An observer inside the building notices that it takes 0.20 s for the tile to pass her window, which has a height of 1.6 \(\mathrm{m}\) . How far above the top of this window is the roof?

One afternoon, a couple walks three-fourths of the way around a circular lake, the radius of which is 1.50 \(\mathrm{km} .\) They start at the west side of the lake and head due south to begin with. (a) What is the distance they travel? (b) What are the magnitude and direction (relative to due east) of the couple's displacement?

In reaching her destination, a backpacker walks with an average velocity of 1.34 \(\mathrm{m} / \mathrm{s}\) , due west. This average velocity results because she hikes for 6.44 \(\mathrm{km}\) with an average velocity of \(2.68 \mathrm{m} / \mathrm{s},\) due west, turns around, and hikes with an average velocity of \(0.447 \mathrm{m} / \mathrm{s},\) due east. How far east did she walk?

A cement block accidentally falls from rest from the ledge of a \(53.0-\mathrm{m}\) -high building. When the block is 14.0 \(\mathrm{m}\) above the ground, a man, 2.00 \(\mathrm{m}\) tall, looks up and notices that the block is directly above him. How much time, at most, does the man have to get out of the way?

A football player, starting from rest at the line of scrimmage, accelerates along a straight line for a time of 1.5 s. Then, during a negligible amount of time, he changes the magnitude of his acceleration to a value of 1.1 \(\mathrm{m} / \mathrm{s}^{2}\) . With this acceleration, he continues in the same direction for another 1.2 \(\mathrm{s}\) , until he reaches a speed of 3.4 \(\mathrm{m} / \mathrm{s}\) . What is the value of his acceleration (assumed to be constant) during the initial 1.5 -s period?
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