Three asteroids of identical mass (M 1.03 x 1012 kg) are orbiting their common c

Question

Three asteroids of identical mass (M 1.03 x 1012 kg) are orbiting their common center of mass in a perfect circle of radius R-75.8 km. a. What is the period of orbit of one of these aster- oids? You are standing on one of the asteroids. You are standing on the side of the asteroid which faces out from the circle. Your goal is to jump up off the asteroid and escape the entire three asteroid system. The radius of the asteroid upon which you are standing is 710 meters b. If on Earth you can jump 1.01 meters, what final speed will you have when you escape the Asteroid cluster? An important aspect of this chapter, and the next several chapters, is that we will have to use all of our Physics 211 concepts wrapped up in the new idea that the new chapter presents. Chapter 13, for example, could be called "All of Ph 211 with Gravity". There is only one new concept and equation for this entire chapter: Newton's Law of Universal Gravitation: All other equations in this chapter can and should be directly derived from this one equation (and others that we are already familiar with). For example, we can derive the equation for Universal Gravitational Potential Energy by using the definition of Work. Concepts covered in PH 211 a. Position, velocity and acceleration in one, two and three dimensions You can derive all kinematics equations by using our aa strategy b. Force and acceleration Fnet mã c. Circular Motion and Centripetal Acceleratioin U2 acentrip T d. Action/Reaction Pairs e. Impulse and Conservation of Momentum f. Work and Conservation of Energy

Explanation / Answer

form the given data

asteroid mass, M = 1.03*10^13 kg

rotating about common center of mass, circular orbit radius R = 75,800 m

a. period of orbit = T

now, let speed of rotation be v

then

v = 2*pi*R/T

also

Mv^2/R = GM^2/4R^2

v^2 = GM/4R

hnce

v = sqroot(GM/R)/2

so, T = 2*pi*R/v = 4*pi*R/sqroot(GM/R)

T = 10005346.7226 s = 2779.262978 hours = 115.80262 days = 0.3170503 years

b. on earth, jump= he = 1.01 m

hence

2*g*h = v^2

he = v^2/2g

now, in the asteroid cluster

initial potential energy at one asteroid = -GmM/2R

hence final speed = u

so from conservation of energy

0.5mu^2 = 0.5mv^2 - GMm/2R

u^2 = 2g*he - GM/2R

u = 4.451030023 m/s

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