Question

A 3.5×1010kg asteroid is heading directly toward the center of the earth at a steady 21 km/s. To save the planet, astronauts strap a giant rocket to the asteroid perpendicular to its direction of travel. The rocket generates 5.0×109N of thrust. The rocket is fired when the asteroid is 4.4×106kmaway 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 300 s before running out of fuel. Is the earth saved?

yes

no

Answer 1

  (
t = D/v = (4.4*10^6 km)/(21 km/sec)
t = 2.095 * 10^5 sec = 58.19 hr = 2.42 days

b.

tan(theta) = 6400 km/(4.4*10^6 km)
tan(theta) = 1.4545*10^-3

theta = 8.33 * 10^-2 degrees

(c)  

v_minimum = 6400 km/(2.7 * 10^5 sec)
v_minimum = 23.704 m/s

Using F = m*a, we can calculate the acceleration of the asteroid due to the rocket's thrust:

5.0*10^9 N = 3.5*10^10 kg * a
a = (5.6*10^9 N)/(3.4*10^10 kg)
a = 1.647 * 10^-1 m/s^2

The transverse velocity after 300 seconds of this acceleration is:

v_transverse = a*t = 1.647 * 10^-1 m/s^2 * 300 s = 49.412 m/s = 4.941*10^-2 km/s

This exceeds the minimum velocity to miss the Earth by a comfortable margin.

The angle through which the asteroid is deflected is given by:

tan(deflection_angle) = v_transverse/(21 km/s)
tan(deflection_angle) = (4.941*10^-2 km/s)/(21 km/s) = 2.352*10^-3
deflection_angle = arctan(2.352*10^-3)
deflection_angle = 0.1347 degrees