A 4.9×10^10kg asteroid is heading directly toward the center of the earth at a s

Question

A 4.9×10^10kg asteroid is heading directly toward the center of the earth at a steady 25 km/s . To save the planet, astronauts strap a giant rocket to the asteroid perpendicular to its direction of travel. The rocket generates 5.0×10^9N of thrust. The rocket is fired when the asteroid is 4.3×10^6kmaway from earth. You can ignore the rotational motion of the earth and asteroid around the sun.

Part A

If the mission fails, how many hours is it until the asteroid impacts the earth?

Express your answer to two significant figures and include the appropriate units.

Part B

The radius of the earth is 6400 km. By what minimum angle must the asteroid be deflected to just miss the earth?

Express your answer to two significant figures and include the appropriate units.

Part C

The rocket fires at full thrust for 400 s before running out of fuel. Is the earth saved?

Explanation / Answer

A) if mission fails then .

asteroid have to travel 4.3 x 10^6 km with speed of 25 km/s

time = distance / speed = (4.3 x 10^^ km ) / (25 km/s) = 172000 s

t (in hr) = 172000/3600 = 47.78 hours

B) tan@ = 6400km/ (4.3 x 10^6 km)

@ = 0.085 deg

C) rockets thrust force = F = 5 x 10^9 N

a = F / ( M) = (5 x 10^9 ) / (4.9e+10 ) = 0.102 m/s^2

speed gain in 400 s.

v = u + at = 0 + (0.102 x 400) = 40.82 m/s

distance traveled in 200s = ut + at^2 /2

d1 = 0 + (0.102 x 200^2 /2 ) = 2040 m

time remains = 172000 - 200 = 171800 s

distance traveled in this time = 40.82 x 171800 = 7012876 m

total distance traveled = 2040 + 7012876 = 7014916 m = 7015.92 km

so earth will be saved.

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