A Rocket Fired Straight Up

Q. A rocket is fired vertically with a speed of 5 km/s from the earth's surface. How far from the earth does the rocket go before returning to the earth? Mass of the earth = 6.0 × 10²⁴ kg; mean radius of the earth = 6.4 × 10⁶ m. Answer in metres. expects 1.6 × 10⁶ m

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The web of concepts

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Q. A rocket is fired vertically with a speed of 5 km/s $v_{0,r} = 5$ km/s from the earth's surface $h_{0,re} = 0$ m. How far from the earth does the rocket go before returning $v_{1,r} = 0$ m/s to the earth? Mass of the earth = 6.0 × 10²⁴ kg $m_{e} = 6.0 \times 10^{24}$ kg; mean radius of the earth = 6.4 × 10⁶ m $R_{e} = 6.4 \times 10^{6}$ m. Answer in metres.

The playground

Inputs

speed

v_{0,r}

5 km/s

mass

m_{e}

6e+24 kg

radius

R_{e}

6400000.00 m

Outputs

height

h_{1,re}

The solution

Solved with GravitationalPotentialEnergy

Given

$v_{0,r} = 5$ km/s
$m_{e} = 6.0 \times 10^{24}$ kg
$R_{e} = 6.4 \times 10^{6}$ m
$h_{0,re} = 0$ m from “from the earth's surface”
$v_{1,r} = 0$ m/s from “before returning”

Find $h_{1,re}$ .

Working

$v_{0,r} = 5$ km/s $= 5000$ m/s
height is measured from the surface(rocket, earth, rising, 1) $h_{0,re} = r_{0,re} - R_{e}$
$0 = r_{0,re} - 6.4 \times 10^{6}$
$r_{0,re} = 6.4 \times 10^{6}$ m
potential of a mass(rocket, earth, rising, 1) $V_{g,0,re} = - \frac{G \cdot m_{e}}{r_{0,re}}$
$V_{g,0,re} = - \frac{6.674 \times 10^{−11} \cdot 6 \times 10^{24}}{6.4 \times 10^{6}}$
$V_{g,0,re} = −6.256875 \times 10^{7}$ J/kg
energy kept in free flight(rocket, earth, rising) $\frac{{v_{1,r}}^{2}}{2} + V_{g,1,re} = \frac{{v_{0,r}}^{2}}{2} + V_{g,0,re}$
$\frac{0^{2}}{2} + V_{g,1,re} = \frac{5000^{2}}{2} + (−6.256875 \times 10^{7})$
$V_{g,1,re} = −5.006875 \times 10^{7}$ J/kg
potential of a mass(rocket, earth, rising, 2) $V_{g,1,re} = - \frac{G \cdot m_{e}}{r_{1,re}}$
$(−5.006875 \times 10^{7}) = - \frac{6.674 \times 10^{−11} \cdot 6 \times 10^{24}}{r_{1,re}}$
$r_{1,re} = 7.99780302 \times 10^{6}$ m
height is measured from the surface(rocket, earth, rising, 2) $h_{1,re} = r_{1,re} - R_{e}$
$h_{1,re} = 7.99780302 \times 10^{6} - 6.4 \times 10^{6}$
$h_{1,re} = 1.598 \times 10^{6}$ m
$h_{1,re} \approx 1.6 \times 10^{6}$ mto 2 significant figures

Conclusion

The rocket climbs until its kinetic energy has all gone into climbing out of the earth's pull; the pull weakens with height, so it rises further than it would under a steady g.

The Ruby

1require_relative "../physics"23question :a_rocket_fired_straight_up do4  kind :numerical5  describes "A rocket is fired vertically with a speed of %{speed} from the earth's surface. %{height} before " \6            "returning to the earth? Mass of the earth = %{mass}; mean radius of the earth = %{radius}. Answer in " \7            "metres."8  given speed: "5 km/s"9  given mass: "6.0e24 kg"10  given radius: "6.4e6 m"11  asks height: "How far from the earth does the rocket go"12end

1require_relative "../questions/a_rocket_fired_straight_up"2require_relative "../science/gravitation/gravitational_potential_energy"34solve :a_rocket_fired_straight_up, using: GravitationalPotentialEnergy do5  objects :rocket, :earth6  rocket.goes(:rising)78  take speed: rocket.initial_speed9  take mass: earth.mass10  take radius: earth.radius_of_the_body1112  deduce_from "from the earth's surface", rocket.height_above_the_surface(from: earth, at: rising.start) => "0 m"13  deduce_from "before returning", rocket.final_speed => "0 m/s"1415  height = rocket.height_above_the_surface(from: earth, at: rising.end)16  answer height:, in: "m", figures: 21718  conclusion "The rocket climbs until its kinetic energy has all gone into climbing out of the earth's pull; " \19             "the pull weakens with height, so it rises further than it would under a steady g."20end

1require_relative "../solutions/a_rocket_fired_straight_up"23play_with :a_rocket_fired_straight_up do4  title "A Rocket Fired Straight Up"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 rocket	$g_{s,r}$	m/s²	—
mass of rocket	$m_{r}$	kg	0 … ∞
radius of the body of rocket	$R_{r}$	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 rocket	$E_{g,gr}$	m/s²	—
separation of ground from rocket	$r_{g,gr}$	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 rocket by ground	$E_{g,rg}$	m/s²	—
separation of rocket from ground	$r_{g,r}$	m	0 … ∞
gravitational intensity of rocket by earth	$E_{g,re}$	m/s²	—
separation of rocket from earth	$r_{g,re}$	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 rocket	$E_{g,er}$	m/s²	—
separation of earth from rocket	$r_{g,er}$	m	0 … ∞
height above the surface of rocket from ground at start of rising	$h_{0,r}$	m	—
distance from the centre of rocket from ground at start of rising	$r_{0,r}$	m	—
height above the surface of rocket from ground at end of rising	$h_{1,r}$	m	—
distance from the centre of rocket from ground at end of rising	$r_{1,r}$	m	—
height above the surface of rocket from earth at start of rising	$h_{0,re}$	m	—
distance from the centre of rocket from earth at start of rising	$r_{0,re}$	m	—
height above the surface of rocket from earth at end of rising	$h_{1,re}$	m	—
distance from the centre of rocket from earth at end of rising	$r_{1,re}$	m	—
gravitational potential of rocket from ground at start of rising	$V_{g,0,r}$	J/kg	—
gravitational potential of rocket from ground at end of rising	$V_{g,1,r}$	J/kg	—
gravitational potential of rocket from earth at start of rising	$V_{g,0,re}$	J/kg	—
gravitational potential of rocket from earth at end of rising	$V_{g,1,re}$	J/kg	—
escape speed of rocket from ground at start of rising	$v_{e,0,r}$	m/s	—
escape speed of rocket from ground at end of rising	$v_{e,1,r}$	m/s	—
escape speed of rocket from earth at start of rising	$v_{e,0,re}$	m/s	—
escape speed of rocket from earth at end of rising	$v_{e,1,re}$	m/s	—
speed of rocket at end of rising	$v_{1,r}$	m/s	0 … ∞
speed of rocket at start of rising	$v_{0,r}$	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,r} = \frac{G \cdot m_{r}}{{R_{r}}^{2}}

gravity at the surface(rocket)

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

gravity at the surface(earth)

E_{g,gr} = \frac{G \cdot m_{r}}{{r_{g,gr}}^{2}}

law of gravitation(ground, rocket)

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

law of gravitation(ground, earth)

E_{g,rg} = \frac{G \cdot m_{g}}{{r_{g,r}}^{2}}

law of gravitation(rocket, ground)

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

law of gravitation(rocket, earth)

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

law of gravitation(earth, ground)

E_{g,er} = \frac{G \cdot m_{r}}{{r_{g,er}}^{2}}

law of gravitation(earth, rocket)

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

height is measured from the surface(rocket, ground, rising, 1)

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

height is measured from the surface(rocket, ground, rising, 2)

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

height is measured from the surface(rocket, earth, rising, 1)

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

height is measured from the surface(rocket, earth, rising, 2)

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

potential of a mass(rocket, ground, rising, 1)

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

potential of a mass(rocket, ground, rising, 2)

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

potential of a mass(rocket, earth, rising, 1)

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

potential of a mass(rocket, earth, rising, 2)

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

escape speed from where it is(rocket, ground, rising, 1)

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

escape speed from where it is(rocket, ground, rising, 2)

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

escape speed from where it is(rocket, earth, rising, 1)

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

escape speed from where it is(rocket, earth, rising, 2)

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

energy kept in free flight(rocket, ground, rising)

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

energy kept in free flight(rocket, earth, rising)