Jet Aircraft’s Kinetic Energy Calculate the kinetic energy of a Boeing 767 airliner that is loaded to its maximum weight of 350,000 lb and travels at 400 mph. Express the kinetic energy in both USCS and SI dimensions. Approach We are tasked with finding the instantaneous kinetic energy of

Question

Jet Aircraft’s Kinetic Energy

Calculate the kinetic energy of a Boeing 767 airliner that is loaded to its maximum weight of 350,000 lb and travels at 400 mph. Express the kinetic energy in both USCS and SI dimensions.
Approach
We are tasked with finding the instantaneous kinetic energy of an airliner travelling at a given velocity and having a given weight. To apply Equation (7.3),

[latex]U_{k}=\frac{1}{2}mv ^2[/latex] (7.3)

we must first determine the aircraft’s mass based on its weight (which is given) and convert the speed to dimensionally consistent units.

Accepted Answer

The aircraft’s mass is

[latex]m=\frac{3.5 imes 10^5 lb}{32.2 ft/s^2} \longleftarrow \left[w=mg ight] [/latex]

[latex]=1.087 imes 10^4\frac{lb.s^2}{ft} [/latex]

[latex]=1.087 imes 10^4 slugs[/latex]

where we applied the definition of the derived unit slug from Equation (3.2).

[latex]1slug=1\frac{lb.s^2}{ft} [/latex] (3.2)

By using the definition of the derived unit mile from Table 3.5,

Table 3.5 Certain Derived Units in the USCS

Quantity	Derived Unit	Abbreviation	Definition
Length	mil	mil	1 mil = 0.001 in.
	inch	in.	1 in. = 0.0833 ft
	mile	mi	1 mi = 5280 ft
Volume	gallon	gal	[latex] 1 gal = 0.1337 ft^3 [/latex]
Mass	slug	slug	[latex]1 slug = 1 (lb . S^2)/ft[/latex]
Mass	pound-mass	lbm	[latex] 1 lbm = 3.1081 imes 10^{-2}(lb s^{2})/ft [/latex]
Force	ounce	oz	1 oz = 0.0625 lb
Force	ton	ton	1 ton = 2000 lb
Torque, or moment of a force	foot-pound	ft · lb	—
Pressure or stress	[latex]pound/inch^2[/latex]	psi	[latex]1 psi = 1 lb/in^2[/latex]
Energy, work, or heat	foot-pound	ft · lb	—
Energy, work, or heat	British thermal unit	Btu	1 Btu = 778.2 ft . lb
Power	horsepower	hp	1 hp = 550 (ft . lb)/s
Temperature	degree Fahrenheit	[latex]°F[/latex]	[latex]°F = °R-459.67[/latex]
Although a change in temperature of [latex] 1°[/latex] Rankine also equals a change of 1 [latex]°F[/latex], numerical , numerical values are converted using the formula.

the aircraft’s speed in dimensionally consistent units is

[latex]v=\left(400\frac{mi}{h} ight)\left(5280\frac{ft}{mi} ight)\left(\frac{1}{3600} \frac{h}{s} ight) [/latex]

[latex]=586.7\left(\frac{\cancel{mi}}{\cancel{h}} ight) (\frac{ft}{\cancel{mi}} )\left(\frac{\cancel{h}}{s} ight) [/latex]

[latex]=586.7\frac{ft}{s} [/latex]

The kinetic energy then becomes

[latex]U_{k}=\frac{1}{2}(1.087 imes 10^4 slugs)\left(586.7\frac{ft}{s} ight)^2 \longleftarrow \left[U_{k}=\frac{1}{2}m v ^2 ight] [/latex]

[latex]=1.871 imes 10^9\left(\frac{slug.ft}{s^2} ight)(ft) [/latex]

[latex]=1.871 imes 10^9 ft.lb[/latex]

Referring to Table 7.1,

Table 7.1
Conversion Factors Between Various Units for Energy and Work in the USCS and SI

(ft . lb)	J	Btu	kW.h
1	1.356	[latex]1.285×10^{-3}[/latex]	[latex]3.766×10^{-7}[/latex]
0.7376	1	[latex]9.478×10^{-4}[/latex]	[latex]2.778×10^{-7}[/latex]
778.2	1055	1	[latex]2.930×10^{-4}[/latex]
[latex]2.655×10^{6}[/latex]	[latex]3.600×10^{6}[/latex]	3413	1

this amount of energy is equivalent to

[latex]U_{k}=( 1.871 imes 10^9 ft.lb ) \left(1.356\frac{J}{ft.lb} ight) [/latex]

[latex]=2.537 imes 10^9(\cancel{ft.lb})\left(\frac{J}{\cancel{ft.lb}} ight)[/latex]

[latex]=2.537 imes 10^9 J[/latex]

[latex]=2.537 GJ[/latex]

in the SI.

Discussion
While manipulating the dimensions in this example, we used the definition of the derived unit slug in the USCS [latex](1 slug = 1 (lb. s^2)/ft) [/latex], and the SI prefix “giga” for the factor of 1 billion. Also, recognize that an airliner’s gravitational potential energy would be significant during flight.

[latex]U_{k}=1.871 imes 10^9 ft . lb[/latex]

[latex]U_{k}=2.537 GJ[/latex]

Question 7.3: Jet Aircraft’s Kinetic Energy Calculate the kinetic energy o...

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