1. A rocket is fired straight up through the atmosphere from the South Pole, bur
ID: 1323430 • Letter: 1
Question
1. A rocket is fired straight up through the atmosphere from the South Pole, burning out at an altitude of 204 km when traveling at 5.90 km/s.
What maximum distance from the launch site does it travel before falling back to Earth?
__________ m/s
1. A rocket is fired straight up through the atmosphere from the South Pole, burning out at an altitude of 204 km when traveling at 5.90 km/s. What maximum distance from the launch site does it travel before falling back to Earth? 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/sExplanation / Answer
1)
At Ux = 0 and Uy = U = 5.9 kmps when reaching H = 240 km above the S pole, it has no X velocity component. So it will fall straight back down and destroy the launch site and everyone around it, including the penguins.
OR
Maybe that's escape velocity and it just keeps on truckin'. Let's test that WAG.
From 1/2 mU^2 = mGM/(R + h)^2 * (R + h); we have U = sqrt(2GM/(R + h)) = sqrt(2*6.67E-11*5.96E24/((6340 + 204)*1000)) = 10993.12336 = 10993 mps. So the 5900 mps is not escape velocity.
So the max distance is the max height above the launch pad. And that's:
Height at any time t = y(t) = h + Uy t - 1/2 g t^2
Distance at any time t = x(t) = Ux t
g = 9.810 m/s^2
Launch height above ground H = 204000.000 meters
Impact (target) elevation y(T) = 0.000 meters
h = H - y(T) = 204000.000 meters
Launch speed U = 5900.000 mps
Launch angle theta = 90.000 degrees
Uy = U sin(theta) = 5900.000 mps
Ux = U cos(theta) = 0.000 mps
For total flight time T, solve the quadratic: 0 = 204,000.000 + 5,900.000 T - 4.905 T^2
Quadratic coefficients: A = 4.905 B = -5900.000 C = -204000.000
calculate Total Flight Time sec
T =
Max Range to Impact meters
x(T) = Ux T
Range at Max Height meters
x(tmax)
Max Height above impact elevation meters
y(tmax) = <=================== ANS
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