/*! 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 4 A fully loaded Boeing \(777-200\... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 4

A fully loaded Boeing \(777-200\) jet transport aircraft weighs 325,000 kg. The pilot brings the 2 engines to full takeoff thrust of \(450 \mathrm{kN}\) cach before releasing the brakes. Neglecting aerodynamic and rolling resistance, estimate the minimum runway length and time needed to reach a takeoff speed of 225 kph. Assume engine thrust remains constant during ground roll.

Short Answer

Expert verified
The minimum runway length required for the Boeing 777-200 to reach a takeoff speed of 225 kph is approximately 708 meters, and it would take about 22.6 seconds to reach this speed.

Step by step solution

01

Calculate the Acceleration

We'll first convert the speed of the plane to meters per second (m/s) from kph by multiplying by \( \frac{1000}{3600} \) because 1 kph= \( \frac{1000}{3600} \) m/s. So, 225 kph = \( 225 \times \frac{1000}{3600} = 62.5 \) m/s. Next, calculate the total force acting on the plane which is the summation of the thrust provided by both engines. So, Total force = Thrust of each engine * Number of engines = \( 2 \times 450,000 = 900,000N \). We can calculate acceleration (a) using the formula \( a = \frac{F}{m} \) where F is the total force and m is the mass of the object. So, \( a = 900,000N / 325,000kg = 2.77 m/s^2 \)
02

Calculate the Distance

Now, let's calculate the runway length needed. To do so, we will apply the formula of motion, \( s = ut + \frac{1}{2}at^2 \), where a is acceleration, t is time, u is initial velocity and s is the distance. Here, the initial speed (u) is 0 because the plane starts from rest, so the formula simplifies to \( s = \frac{1}{2}at^2 \). However, we don't know the time, so we need to rearrange the linear motion formula \( v = u + at \), to find time \( t = (v-u)/a \). Again as u is 0, it will simplify to \( t = v/a \). Replace these values of v and a to get, \( t = 62.5/2.77 \approx 22.6s \). And then place the values of a and t to calculate s. So, \( s = \frac{1}{2} \times 2.77 \times (22.6)^2 = 708 \) meters.
03

Confirm the Results

We calculated the time and the runway length required to reach the takeoff speed for airplane. The time to reach the takeoff speed is approximately 22.6 seconds and the runway length used is approximately 708 meters. These calculations assume constant engine thrust and neglect air and rolling resistances, meaning actual values may vary.

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