a huge cannon is assembled on an airless planet. The planet has a raduis 5.00*10

Question

a huge cannon is assembled on an airless planet. The planet has a raduis 5.00*106m and a mass of 4.6*1024kg. the cannon fires a projectile up at 4000 m/s.
1) To what maximum height does the projectile reach above the surface?
2) The projectile reaches its maximum height and then starts to fall back to the planet. what is its speed as it passes the 500 km mark above the surface?
3) show that the escape volecity formula is VESCAPE =sqrt(2GM/R)
4) what is the escape velocity of the planet ?
5) If a 2000 kg satellite were circularly orbiting this planet at height (above the surface) of 1000 km what would its total energy be?
6) If a 2000 kg satellite were circularly orbiting this planet at height (above the surface) of 1000 km what would its speed be?
7) If a 2000 kg satellite were circularly orbiting this planet at height (above the surface) of 1000 km how much energy would the thrusters need to supply to move it to an orbit 2000 km above the surface?

MUST SHOW WORK AND STEPS

Explanation / Answer

1)

first we must find the acceleration:

g = G M/R^2 = (6.67e-11)*(4.6e24)/(5e6)y2 = 12.273 m/s2

hmax = v^2/(2 g) = (4000*4000)/(2*12.273) = 6.5184e5 m = 6.52 x 10^5 m

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2)

v = sqrt(2 g d) = sqrt(2*12.273*(6.5184e5 - 500e3)) = 1930 m/s

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3)

according the conservation of energy:

(1/2) m (vescape)^2 = G M m/R

cancel m:

==> (1/2) (vescape)^2 = G M/R

==> vescape = sqrt(2 G M/R)

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4)

vescape = sqrt(2 G M/R) = sqrt(2*(6.67e-11)*(4.6e24)/(5e6)) = 11078 m/s = 1.11 x 10^4 m/s

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5)

first we find it's velocity:

m v^2/r = G m M/r^2

==> v = sqrt(G M/r) = sqrt((6.67e-11)*(4.6e24)/(5e6+1000e3)) = 7151 m/s

total energy:

E = -G M m/r + (1/2) m v^2

==> E = -(6.67e-11)*(4.6e24)*(2000)/(5e6+1000e3) + (1/2)*(2000)*(7151*7151)

==> E = -5.1137 x 10^10 J

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6)

m v^2/r = G m M/r^2

==> v = sqrt(G M/r) = sqrt((6.67e-11)*(4.6e24)/(5e6+1000e3)) = 7151 m/s

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7)

E = G m M (1/2ri - 1/2rf)

==> E = (6.67e-11)*(4.6e24)*(2000) * (1/(2*(5e6+1000e3)) - 1/(2*(5e6+2000e3)))

==> E = 7.31 x 10^9 J

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