_______ m/s A rocket is fired straight up through the atmosphere from the South
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Question
_______ m/s
A rocket is fired straight up through the atmosphere from the South Pole, burning out at an altitude of 247 km when traveling at 4.90 km/s. What maximum distance from the launch site does it travel before falling back to Earth?
____ m
1. A space probe is fired as a projectile from the Earth's surface with an initial speed of 1.58 x 10^4 m/s. What will its speed be when it is very far from the Earth? Ignore atmospheric friction and the rotation of the Earth. _______ m/s A rocket is fired straight up through the atmosphere from the South Pole, burning out at an altitude of 247 km when traveling at 4.90 km/s. What maximum distance from the launch site does it travel before falling back to Earth? ____ mExplanation / Answer
Initial speed of the space probe is, v0 = 1.58 * 104 m/s
Mass of the Earth is, M = 5.972 * 1024 kg
Radius of the earth is R = 6.371 * 106 m
Universal gravitational constant is, G = 6.67384 * 10-11 m3/kg.s2
Initial energy of the space probe is,
Ei = (1/2) m v02 - G m M / R
where m is the mass of the space probe.
Final energy of the space probe is,
Ef = (1/2) m v2 - G m M / r
where r >> R and we take it to be infinite.
From the law of conservation of energy,
Ef = Ei
(1/2) m v2 - G m M / r = (1/2) m v02 - G m M / R
v2 = 2 G M / r + v02 - 2 G M / R
v2 = 0 + 1.58 * 104 * 1.58 * 104 - 6.67384 * 10-11 * 5.972 * 1024 / 6.371 * 106
v2 = 1.870812612 * 108 m2/s2
v = 1.367776521 * 104 m/s = 13677.765 m/s is the speed of the space probe when it is very far away from the Earth.
A rocket is fired straight up through the atmosphere from the South Pole, burning out at an altitude of 247 km when traveling at 4.90 km/s.
Initial speed of the space probe is, v0 = 4.9 * 103 m/s
Mass of the Earth is, M = 5.972 * 1024 kg
Radius of the earth is R = 6.371 * 106 m
Universal gravitational constant is, G = 6.67384 * 10-11 m3/kg.s2
Since the rocket is at a distance of 247 km from the Earth's surface, g will have a slightly different value than the one we use for objects on/near the surface of the earth.
g = GM / r2
Here, r = R (Earth radius) + the altitude height
r = 6.371 * 106 + 0.247 * 106
r = 6.618 * 106 m
Thus,
g = 6.67384 * 10-11 * 5.972 * 1024 / (6.618 * 106 * 6.618 * 106)
g = 9.1 m/s2
Maximum height (maximum distance from the launch site) of this rocket can reach is,
H = v02 / 2g
H = 4.9 * 103 * 4.9 * 103 / (2 * 9.1)
H = 1.319230769 * 106 m = 1319230.769 m
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