Thrice the Escape Speed

Q. The escape speed of a projectile on the earth's surface is 11.2 km/s. A body is projected out with thrice this speed. What is the speed of the body far away from the earth? Ignore the presence of the sun and other planets. Answer in km/s. expects 31.7 km/s

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The web of concepts

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Q. The escape speed of a projectile on the earth's surface is 11.2 km/s $v_{e,0,be} = 11.2$ km/s. A body is projected out with thrice this speed $v_{0,b} = 3 \cdot v_{e,0,be}$ . What is the speed of the body far away from the earth $V_{g,1,be} = 0$ J/kg? Ignore the presence of the sun and other planets. Answer in km/s.

The playground

Inputs

escape

v_{e,0,be}

11.2 km/s

Outputs

speed

v_{1,b}

The solution

Solved with GravitationalPotentialEnergy

Given

$v_{e,0,be} = 11.2$ km/s
$V_{g,1,be} = 0$ J/kg from “far away from the earth”
$v_{0,b} = 3 \cdot v_{e,0,be}$ from “thrice this speed”

Find $v_{1,b}$ .

Working

$v_{e,0,be} = 11.2$ km/s $= 11200$ m/s
thrice this speed $v_{0,b} = 3 \cdot v_{e,0,be}$
$v_{0,b} = 3 \cdot 11200$
$v_{0,b} = 33600$ m/s
escape speed from where it is(body, earth, away, 1) ${v_{e,0,be}}^{2} = -2 \cdot V_{g,0,be}$
$11200^{2} = -2 \cdot V_{g,0,be}$
$V_{g,0,be} = −6.272 \times 10^{7}$ J/kg
energy kept in free flight(body, earth, away) $\frac{{v_{1,b}}^{2}}{2} + V_{g,1,be} = \frac{{v_{0,b}}^{2}}{2} + V_{g,0,be}$
$\frac{{v_{1,b}}^{2}}{2} + 0 = \frac{33600^{2}}{2} + (−6.272 \times 10^{7})$
$v_{1,b} = 31.68$ km/s
$v_{1,b} = 31678.3838$ m/s $= 31.6783838$ km/s
$v_{1,b} \approx 31.7$ km/sto 3 significant figures

Conclusion

Escaping uses up exactly the kinetic energy the escape speed carries, and what is left over is kept far away, so the squares of the speeds subtract.

The Ruby

1require_relative "../physics"23question :thrice_the_escape_speed do4  kind :numerical5  describes "The escape speed of a projectile on the earth's surface is %{escape}. A body is projected out with " \6            "thrice this speed. %{speed}? Ignore the presence of the sun and other planets. Answer in km/s."7  given escape: "11.2 km/s"8  asks speed: "What is the speed of the body far away from the earth"9end

1require_relative "../questions/thrice_the_escape_speed"2require_relative "../science/gravitation/gravitational_potential_energy"34solve :thrice_the_escape_speed, using: GravitationalPotentialEnergy do5  objects :body, :earth6  body.goes(:away)78  take escape: body.escape_speed(from: earth, at: away.start)910  deduce_from "thrice this speed", body.initial_speed => 3 * body.escape_speed(from: earth, at: away.start)11  deduce_from "far away from the earth", body.gravitational_potential(from: earth, at: away.end) => "0 J/kg"1213  speed = body.final_speed14  answer speed:, in: "km/s", figures: 31516  conclusion "Escaping uses up exactly the kinetic energy the escape speed carries, and what is left over is " \17             "kept far away, so the squares of the speeds subtract."18end

1require_relative "../solutions/thrice_the_escape_speed"23play_with :thrice_the_escape_speed do4  title "Thrice the Escape Speed"5end

The laws

quantity	written	units	within	from
gravitational constant	$G$	N m² kg⁻²	—	UniversalGravitation
mass	$m$	kg	0 … ∞	UniversalGravitation
radius of the body	$R$	m	0 … ∞	UniversalGravitation
surface gravity	$g_{s}$	m/s²	—	UniversalGravitation
separation	$r_{g}$	m	0 … ∞	UniversalGravitation
gravitational intensity	$E_{g}$	m/s²	—	UniversalGravitation
distance from the centre	$r$	m	—	GravitationalPotentialEnergy
height above the surface	$h$	m	—	GravitationalPotentialEnergy
gravitational potential	$V_{g}$	J/kg	—	GravitationalPotentialEnergy
escape speed	$v_{e}$	m/s	—	GravitationalPotentialEnergy
speed	$v$	m/s	0 … ∞	GravitationalPotentialEnergy
surface gravity of ground	$g_{s,g}$	m/s²	—
mass of ground	$m_{g}$	kg	0 … ∞
radius of the body of ground	$R_{g}$	m	0 … ∞
surface gravity of body	$g_{s,b}$	m/s²	—
mass of body	$m_{b}$	kg	0 … ∞
radius of the body of body	$R_{b}$	m	0 … ∞
surface gravity of earth	$g_{s,e}$	m/s²	—
mass of earth	$m_{e}$	kg	0 … ∞
radius of the body of earth	$R_{e}$	m	0 … ∞
gravitational intensity of ground by body	$E_{g,gb}$	m/s²	—
separation of ground from body	$r_{g,gb}$	m	0 … ∞
gravitational intensity of ground by earth	$E_{g,ge}$	m/s²	—
separation of ground from earth	$r_{g,ge}$	m	0 … ∞
gravitational intensity of body by ground	$E_{g,bg}$	m/s²	—
separation of body from ground	$r_{g,b}$	m	0 … ∞
gravitational intensity of body by earth	$E_{g,be}$	m/s²	—
separation of body from earth	$r_{g,be}$	m	0 … ∞
gravitational intensity of earth by ground	$E_{g,eg}$	m/s²	—
separation of earth from ground	$r_{g,e}$	m	0 … ∞
gravitational intensity of earth by body	$E_{g,eb}$	m/s²	—
separation of earth from body	$r_{g,eb}$	m	0 … ∞
height above the surface of body from ground at start of away	$h_{0,b}$	m	—
distance from the centre of body from ground at start of away	$r_{0,b}$	m	—
height above the surface of body from ground at end of away	$h_{1,b}$	m	—
distance from the centre of body from ground at end of away	$r_{1,b}$	m	—
height above the surface of body from earth at start of away	$h_{0,be}$	m	—
distance from the centre of body from earth at start of away	$r_{0,be}$	m	—
height above the surface of body from earth at end of away	$h_{1,be}$	m	—
distance from the centre of body from earth at end of away	$r_{1,be}$	m	—
gravitational potential of body from ground at start of away	$V_{g,0,b}$	J/kg	—
gravitational potential of body from ground at end of away	$V_{g,1,b}$	J/kg	—
gravitational potential of body from earth at start of away	$V_{g,0,be}$	J/kg	—
gravitational potential of body from earth at end of away	$V_{g,1,be}$	J/kg	—
escape speed of body from ground at start of away	$v_{e,0,b}$	m/s	—
escape speed of body from ground at end of away	$v_{e,1,b}$	m/s	—
escape speed of body from earth at start of away	$v_{e,0,be}$	m/s	—
escape speed of body from earth at end of away	$v_{e,1,be}$	m/s	—
speed of body at end of away	$v_{1,b}$	m/s	0 … ∞
speed of body at start of away	$v_{0,b}$	m/s	0 … ∞

g_{s,b} = \frac{G \cdot m_{b}}{{R_{b}}^{2}}

gravity at the surface

E_{g,oo} = \frac{G \cdot m_{o}}{{r_{g,oo}}^{2}}

law of gravitation

g_{s,g} = \frac{G \cdot m_{g}}{{R_{g}}^{2}}

gravity at the surface(ground)

g_{s,b} = \frac{G \cdot m_{b}}{{R_{b}}^{2}}

gravity at the surface(body)

g_{s,e} = \frac{G \cdot m_{e}}{{R_{e}}^{2}}

gravity at the surface(earth)

E_{g,gb} = \frac{G \cdot m_{b}}{{r_{g,gb}}^{2}}

law of gravitation(ground, body)

E_{g,ge} = \frac{G \cdot m_{e}}{{r_{g,ge}}^{2}}

law of gravitation(ground, earth)

E_{g,bg} = \frac{G \cdot m_{g}}{{r_{g,b}}^{2}}

law of gravitation(body, ground)

E_{g,be} = \frac{G \cdot m_{e}}{{r_{g,be}}^{2}}

law of gravitation(body, earth)

E_{g,eg} = \frac{G \cdot m_{g}}{{r_{g,e}}^{2}}

law of gravitation(earth, ground)

E_{g,eb} = \frac{G \cdot m_{b}}{{r_{g,eb}}^{2}}

law of gravitation(earth, body)

h_{0,b} = r_{0,b} - R_{g}

height is measured from the surface(body, ground, away, 1)

h_{1,b} = r_{1,b} - R_{g}

height is measured from the surface(body, ground, away, 2)

h_{0,be} = r_{0,be} - R_{e}

height is measured from the surface(body, earth, away, 1)

h_{1,be} = r_{1,be} - R_{e}

height is measured from the surface(body, earth, away, 2)

V_{g,0,b} = - \frac{G \cdot m_{g}}{r_{0,b}}

potential of a mass(body, ground, away, 1)

V_{g,1,b} = - \frac{G \cdot m_{g}}{r_{1,b}}

potential of a mass(body, ground, away, 2)

V_{g,0,be} = - \frac{G \cdot m_{e}}{r_{0,be}}

potential of a mass(body, earth, away, 1)

V_{g,1,be} = - \frac{G \cdot m_{e}}{r_{1,be}}

potential of a mass(body, earth, away, 2)

{v_{e,0,b}}^{2} = -2 \cdot V_{g,0,b}

escape speed from where it is(body, ground, away, 1)

{v_{e,1,b}}^{2} = -2 \cdot V_{g,1,b}

escape speed from where it is(body, ground, away, 2)

{v_{e,0,be}}^{2} = -2 \cdot V_{g,0,be}

escape speed from where it is(body, earth, away, 1)

{v_{e,1,be}}^{2} = -2 \cdot V_{g,1,be}

escape speed from where it is(body, earth, away, 2)

\frac{{v_{1,b}}^{2}}{2} + V_{g,1,b} = \frac{{v_{0,b}}^{2}}{2} + V_{g,0,b}

energy kept in free flight(body, ground, away)

\frac{{v_{1,b}}^{2}}{2} + V_{g,1,be} = \frac{{v_{0,b}}^{2}}{2} + V_{g,0,be}

energy kept in free flight(body, earth, away)

v_{0,b} = 3 \cdot v_{e,0,be}

thrice this speed