The pilot of a jet transport brings the engines to full...

The pilot of a jet transport brings the engines to full takeoff power before releasing the brakes as the aircraft is standing on the runway. The jet thrust remains constant, and the aircraft has a nearconstant acceleration of $0.4 \mathrm{g}$. If the takeoff speed is $200 \mathrm{km} / \mathrm{h},$ calculate the distance $s$ and time $t$ from rest to takeoff.

Key Concepts

Uniform Acceleration

This concept refers to a situation where an object's acceleration remains constant throughout its motion. In problems like this, where the aircraft accelerates at a near-constant 0.4g, the constant acceleration permits the use of specific kinematic equations to predict motion parameters such as velocity, time, and distance covered.

Kinematic Equations

Kinematic equations are mathematical formulas that relate displacement, velocity, acceleration, and time under constant acceleration conditions. They enable one to calculate unknown quantities when certain variables are provided, such as using s = (1/2)at² for displacement and v = at for velocity when starting from rest.

Unit Conversion

Unit conversion is the process of converting measurements from one unit to another to ensure consistency in calculations. In this problem, converting takeoff speed from kilometers per hour to meters per second is crucial because the standard units in kinematic equations use meters, seconds, and meters per second.

Gravitational Acceleration

This concept involves the standard acceleration due to gravity, denoted as g, which is approximately 9.8 m/s². Expressing the aircraft's acceleration as a fraction of g (0.4g) is common in aeronautics, and understanding this relationship is essential for scaling the acceleration used in the kinematic equations.

Transcript

00:01 In this problem, an airplane accelerates from 0 to 200 kilometers per hour over some distance s under some constant acceleration or near constant acceleration, 0 .4g.

00:24 All right, so g in this case is 9 .81 meters per second, so this is the same as 0 .4 times.

00:31 9 .81 meters per second.

00:33 Our question asks, what is the distance s that we need to accelerate all the way to the takeoff speed? and what is the distance t that we need to travel over this distance s? so first, let's convert 200 kilometers per hour to 200 kilometers per hour to meters per second, because gravity is measured in meters per second in this case.

01:02 So we can multiply this by a few conversion factors.

01:06 1 ,000 meters on top, 1 kilometer on bottom, so the kilometers cancel, and then another conversion factor 1 hour on top, and then 3 ,600 seconds on bottom.

01:25 So that be, hours cancel.

01:28 This gives you something like 55 .556 seconds, and it's like 55 .555 .5 repeating, but we can just keep this many decimals that'll make our answer accurate for the next step.

01:46 So for motion in one dimension under constant acceleration, our final velocity is given by the initial velocity plus acceleration times time.

01:58 All right.

01:58 So in this case, initial velocity, let's label these vi and vf.

02:03 In this case, initial velocity is zero.

02:06 So we don't need to include it and we get something like 200 oh not 200 kilometers per hour but we just wrote this out let's just call that v final and we know that this is v final because i don't want to rewrite all that right now but v final 55 .556 seconds is equal to 0 .4 times 9 .81 meters per second square that's the acceleration in this case times t so then we have t is given by and i'll actually rewrite it this time 55 .556 meters per second.

02:46 See why do i have seconds here? that's wrong.

02:53 Our thing is measured meters per second.

02:55 We have meters on top, seconds on bottom.

02:57 I can't believe i made that mistake.

02:59 Anyway, meters per second on top for this point.

03:03 And then on the bottom, 0 .4 times 9 .81 meters per second squared.

03:10 And this thing, the meters cancel out.

03:14 And one power of the seconds cancels out so our units is seconds...