This is an open book/notes test to be taken home. You are taking the test under the honor
system, that is you are doing your own work, not helping anybody and not receiving help from
anybody; your signature is your pledge. No signature, no grading!
 

Your name.........................................Your pledge signature...............................................

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1.- A particle of mass m lies in the middle (point A) of a hollow tube of length 2b and mass M.
The tube is closed at both ends and lies on a smooth table(no friction). The coefficient of restitution between m and M is e.  Let m be given an initial velocity V₀ by an external agent and along the tube.
a) find velocities of m and M after the first impact.
b) obtain the energy loss during first impact; call it DT.
c) calculate the time required for m to arrive back at A travelling in the original direction.
d) determines the velocities relative to the center of mass before and after the first collision.
e) what is the energy lost at the first impact in terms of the reference system with its origin attached to the
center of mass (i.e. the CM system).
f) how far has the center of mass travelled during the time it takes m  to arrive back at A, travelling in the
original direction as in c).

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2. A beam of alpha particles (⁴ He nuclei) strikes the atoms of a helium target. For the sake of
this problem, neglect the mass of the 2 electrons, and thus assume that the beam and target particles have
the same mass. If the beam energy is 4 MeV, and the projectile is scattered elastically by 30⁰ in the lab
frame:
a) what are the laboratory kinetic energies of the scattered and recoiling particles?
b) what is the CM energy of the collision?
c) what are the CM kinetic energies of the two particles?
d) what is the CM scattering angle?
e) at what lab angle does the target helium atom recoil.
f) what the CM and lab Rutherford cross section for this scattering situation?
g) Suppose the target is liquid helium with density 0.125 gcm^-2 and 8 cm long, and that the detector has a
 1 cm² sensitive area located 1 m from the target (assume point-like target to obtain detector solid angle);
what fraction of the incident alpha particles hit the detector? How large must the incident beam flux
be if the detector registers one scatter per second?
hint: you may first assume that the incident flux is 1 cm^-2 s^-1 , and calculate the number of target centers in
the beam as given by (density x length x Avrogadro number), to obtain the fraction of incident particles
hitting the detector.

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3.- A projectile is fired due West from a point at a Northern  latitude l on Earth with a velocity of
magnitude V₀ at an angle of inclination to the horizon a.
a) Derive (showing all the steps) the relation for the lateral displacement, d, of the projectile when it strikes
the earth which rotates  with angular velocity w, neglecting the change of range (and therefore of flight time)
due to the Coriolis force .
b) in  what direction is the displacement?
c) assuming now that V₀ =2000 ms^-1 and a=l=45⁰ , what is the numerical value of this displacement?
d) what fraction of the range does it represent?
e) show numerically that it is a good approximation to neglect the curvature of the earth surface.

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                                                                            The End