A Rocket Fired Straight Up

solved GravitationalPotentialEnergy

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.

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}

How it's solved

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 web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

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

solution.rb

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

playground.rb

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 it drew on

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)