DOC PREVIEW
UIUC PHYS 325 - HW4

This preview shows page 1 out of 2 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 2 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 2 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Physics 325 HW 4A due 1:00 pm TUESDAY February 24, 2015 name _________________1. (15 points) A satellite is in a circular orbit around the earth; the orbit has radius r. (from whichyou can determine its speed) It applies its thrusters and suddenly increases its speed (maintaining thedirection of its motion) by a factor χ > 1. •What must χ be if the final orbit is to be parabolic?•Given that χ is actually less than this, so that the final orbit is elliptical, what is the apogee of that orbit? •What is the perigee?Hint: what are the α and ε and e and l of the new orbit in terms of χ ?2. 30 points. A comet of (infinitesimal mass µ) and speed vapproaches the sun, of mass M. At t = -∞ its trajectory is characterizedby its "impact parameter" b. ( b describes by how much it would missthe sun if there were no gravity). Answers should be given as functionsof v, b, M, and G.•What is the angular momentum per mass l of the comet in terms of vand b? ( this can be established at t= -∞ while the comet is a largedistance away, but it's best not to think of r2dφ/dt, but rather of l= r x v; so | l | = rv times the sine of the angle between them. The result is asimple function of v and b.)•What is the energy e per mass of the comet? (as with l, this is bestestablished while the comet is a large distance away where Φ is zero. )•What kind of orbit is it in? Elliptical? Hyperbolic? Parabolic?Long after passing by the sun, it is observed to be departing the sun's vicinity along an (asymptotically)straight line that, extended back towards the sun, misses by a distance b'.•What speed v' does it have as time goes to t=+∞ ? (Use Energy conservation)•What is the exit parameter b' that characterizes its trajectory at t = +∞ ? (Use L conservation)•How close rmin did it get to the sun? Numerically evaluate rmin/b for the case v2b = GM•Through what angle γ (see figure for definition) does the original straight-line trajectory getdiverted?Evaluate the numerical value of γ for the case v2b = GM.To answer these questions you may find it useful to recall the formular(φ) = ( l2/GM ) [ 1 + { 1 + (2e l 2/ G2M2) }1/2 cos(φ) ]-1 = α / [ 1 + ε cos φ ]3. 15 points (cf miniquiz 8) A force field in 2-d rF = −k rˆr ( and potential U = kr2/2) acts on aparticle of mass m.As a function of r, m and k, what is the period of a circular orbit of radius r?As a function of r, m and k, what is the angular momentum of a circular orbit of radius r?As a function of r, m and k, what is the energy of a circular orbit of radius r?The particle is now in a non-circular orbit. In terms of the particle's angular momentum L andenergy E,What is its closest approach to the origin?What is its furthest distance from the origin?Physics 325 HW 4B due 2:00 pm TUESDAY February 24, 2015 name _________________1. 30 points A satellite is in a circular orbit (labeled 1) around the earth (mass M) at radius R withspeed vinitial= √GM/R. We wish it to climb to a higher circular orbit, at radius R' > R with speedvfinal= √GM/R'. We will do this by means of an elliptical Hohmann transfer orbit . It applies animpulse (a change of speed Δv) at time zero, which puts it into the transfer orbit (labeled 2). Onarriving at the new distance R it is at the apogee of the transfer orbit. It then applies an additionalimpulse Δv', increasing its speed again, to enter the final circular orbit (labeled 3).2 15 points Two black holes, of masses 2 Msolar and 1 Msolar respectively, are in circular orbits aroundeach other. Their separation is d = 10,000 km. They each have a speed relative to the joint center ofmass (indicated in the figure by a *).•What is the ratio of their speeds relative to the center of mass? Which is moving faster?(hint : think about total system momentum)•What is the period of their orbit in seconds ?•Where is each black hole at a time 1/4 period later than thetime in the figure? Indicate these positions in the figure. P.S. Msolar= 2 x 1030 kg G = 6.67 x 10-11 N m2/ kg2.3. 15 points A spaceship is in a circular orbit above a black hole ofone solar mass. The orbit has a radius D. • In terms of G, Msolar andD, what is the spaceship's acceleration at this moment? • What isthe period P of its orbit? • Evaluate P numerically (you may use acalculator) for D = 5000 km, Msolar = 2 x 1030 kg; G = 6.67 x 10-11 Nm2/ kg2• What is the tidal effect on a particle floating in the spaceship? Ie.,given a particle a distance h to the right of the spaceship's center of mass (with h << D), what is theparticle's acceleration relative to the ship's center of mass? Evaluate this numerically (you may use acalculator) for D = 5000 km, h = 1 meter, Msolar = 2 x 1030 kg; G = 6.67 x 10-11 N m2/ kg2What is the angular momentum l and energy e (permass) of the transfer orbit? [ This is readily found fromyour knowledge of R and R' and knowing the transferorbit's formula for r(φ); see below. ]What are the needed instantaneous changes of velocityΔv and Δv' ?All answers should be in terms of M, G, R and R'.To answer these questions you may find it useful torecall the formular(φ) = ( l2/GM ) [ 1 + { 1 + (2e l 2/ G2M2) }1/2 cos(φ) ]-1= α / [ 1 + ε cos φ ] Hint: what are α and ε in termsof R and R' ? Note that e must be


View Full Document

UIUC PHYS 325 - HW4

Download HW4
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view HW4 and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view HW4 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?