Answer:
a. t = 69.4 hr = 2.89 days
b. theta = 91.67*10^-3 degrees
c. deflection_angle = 0.134 degrees
Explanation:
a).
The asteroid impacts the earth in t
t = x/v = (4.0*10^6 km)/(16 km/sec)
t = 2.5 * 10^5 sec
t = 69.4 hr = 2.89 days
b).
tan(theta) = 6400 km/(4.0*10^6 km)
tan(theta) = 1.6*10^-3
theta = arctan(1.6*10^-3)
theta = 1.6*10^-3 radians (for small angles, tan(theta) ~= theta)
theta = 91.67*10^-3 degrees
c).
v_minimum = 6400 km/(2.5 * 10^5 sec)
v_minimum = 25.6 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 = 4.0*10^10 kg * a
a = (5.0*10^9 N)/(4.0*10^10 kg)
a = 0.125 m/s^2
The transverse velocity after 300 seconds of this acceleration is:
v_transverse = a*t = 0.125 m/s^2 * 300 s
v_transverse = 37.5 m/s = 37.5*10^-3 km/s
tan(deflection_angle) = v_transverse/(20 km/s)
tan(deflection_angle) = (37.5*10^-3 km/s)/(16 km/s) = 2.34^-3
deflection_angle = arctan(2.34*10^-3)
deflection_angle = 2.34*10^-3 radians = 0.134 degrees
v_transverse/(16 km/s) > (6400km)/(5.0*10^6 km)
(note that the right hand side if this inequality is tan(theta) calculated above)
v_transverse > 23.704 m/
We have that for the Question "Part A: If the mission fails, how many hours is it until the asteroid impacts the earth?
Part B: The radius of the earth is 6400 km. By what minimum angle must the asteroid be deflected to just miss the earth?
Part C: What is the actual angle of deflection if the rocket fires at full thrust for 300 s before running out of fuel?",it can be said that
are appropriate
From the question we are told
A 4.0×1010kg asteroid is heading directly toward the center of the earth at a steady 16 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 4.0×106km away from earth. You can ignore the earth’s gravitational force on the asteroid and their rotation about the sun.
Part A: If the mission fails, how many hours is it until the asteroid impacts the earth?
Part B: The radius of the earth is 6400 km. By what minimum angle must the asteroid be deflected to just miss the earth?
Part C: What is the actual angle of deflection if the rocket fires at full thrust for 300 s before running out of fuel?
a)
Generally the equation for the time is mathematically given as
[tex]t=\frac{d}{v}\\\\t=\frac{4*10^6}{16km/s}[/tex]
t=68hr
b)
Generally the equation for the angle is mathematically given as
[tex]tan\theta=\frac{R}{4*10^6}\\\\tan\theta=\frac{6400}{4*10^6}[/tex]
[tex]\theta=0.09[/tex]
c)
Generally the equation for the acceleration is mathematically given as
[tex]a=\frac{Force}{asteroid\ mass} \\\\ a=\frac{5*10^{9}}{4*10^{10}} \\\\ a=0.125[/tex]
[tex]v_t=at\\\v_t=(0.125)(300)\\\\v_t=38m/s[/tex]
Since
[tex]tan\theta=\frac{v_t}{v}\\\\tan\theta=\frac{37}{16}\\\\\theta=tan^{-1}\frac{37.5}{16*1000}\\\\[/tex]
[tex]\theta=0.134 \textdegree[/tex]
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