MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Physics
8.01
Problem Solving Session 6 Collisions
Section
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Table
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Group Members
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Hand in one set of solutions per group.
Reference Frames
There are many problems involving conservation of momentum in which the
analysis is easier depending on your choice of reference frame. In the notes that follow
we describe how velocities of objects are related in different reference frames and in
particular describe the center of mass reference frame.
Let R be the vector from the origin of a reference frame S to the origin of a
second reference frame S  . Denote the position vector of particle i with respect to the
origin of reference frame S by ri and similarly, denote the position vector of particle i
with respect to the origin of reference frame S  by ri (Figure 1).
Figure 1 Position vector of i th particle in two reference frames.
The position vectors are related by
ri  ri  R .
The relative velocity between the two reference frames is given by
(1)
V  dR .
dt
(2)
When the relative velocity between the two reference frames is constant then the relative
acceleration between the two reference frames is zero,
A  dV  0 ,
dt
(3)
When Eq. (3) is satisfied, the reference frames S and S  are called relatively inertial
reference frames.
Suppose the i th particle in Figure 1 is moving; then observers in different
reference frames will measure different velocities. Denote the velocity of i th particle in
r
r
frame S by v i  dri / dt , and the velocity of the same particle in frame S  by
r
r
v i  dr  / dt . Since the derivative of the position is velocity, the velocities of the particles
in two different reference frames are related according to
vi  vi  V .
(4)
This relation is called the law of addition of velocities and is the primary tool that we will
use when we describe velocities in different reference frames.
r
Example: Suppose a cart has velocity V  Vφ
i with respect to the ground. Let S denote a
reference frame fixed to the ground and let S  denote the reference frame moving with
r
velocity V  Vφ
i with respect to the ground. In S  the cart is at rest. Suppose aball is
r
thrown from the cart with velocity u  uφ
i relative to the cart. Then the velocity of the
ball relative to the ground reference frame S is
r
r r
u  u  V  uφ
i  Vφ
i  (V  u)φ
i.
(5)
Center of Mass Reference Frame
r
Consider a system of N particles. Let R cm be the vector from the origin of a
r
reference frame S to the center of mass of the system of particles. Let r j denote the
position of the j th particle. Recall the definition of the position of the center of mass
j N
r
R cm 
r
mr
j 1
j j
j N
m
j 1
j

1
mT
j N
r
mr
j 1
j j
(6)
where mT 
j N
m
j 1
j
is the mass of the system. The velocity of the center of mass is given
by
r
1
Vcm 
mT
j N
r
m v
j 1
j
j
(7)
r
where v j is the velocity of the j th particle with respect to reference frame S . Choose
the center of mass to be the origin of reference frame S cm , called the center of mass
reference frame. Denote the position vector of particle i with respect to origin of
r
reference frame S by ri and similarly, denote the position vector of particle i with
r
respect to origin of reference frame S cm by rcm,i (Figure 2).
Figure 2 Position vector of ith particle in the center of mass reference frame.
The position vector of particle i in the center of mass frame is then given by
r
r r
rcm,i  ri  R cm .
(8)
The velocity of particle i in the center of mass reference frame is then given by
r
r
r
v cm,i  v i  Vcm .
(9)
IC_W07D3-1 Group Problem: Railroad Gun
A railroad gun of mass m1 fires a shell of mass m2 at an angle of  with respect to
the horizontal as measured in a reference frame moving with the final velocity of the
r
gun v1 f after the shell has been fired. After the firing is complete, the final speed of
r
r
the projectile relative to the gun (muzzle velocity) is v 2 (with speed u  v 2 ). The
r
gun recoils with speed v f  v1 f and the instant the projectile leaves the gun, it
makes an angle  with respect to the ground.
r
a) What is the velocity, v 2 , of the projectile with respect to the ground?
b) Find v f and  in terms of m1 , m2 , u , and  .
Problem 2 This problem is on pset 7 and is a difficult problem so we will get you
started today.
A thin target of lithium is bombarded by helium nuclei of energy E0 . The lithium nuclei
are initially at rest in the target but are essentially unbound. When a helium nucleus
enters a lithium nucleus, a nuclear reaction can occur in which the compound nucleus
splits apart into a boron nucleus and a neutron. The collision is inelastic, and the final
kinetic energy is less than E0 by 2.8 MeV . ( 1 MeV=106 eV=1.6 10-13 J ). The relative
masses of the particles are: helium, mass 4 ; lithium, mass 7 ; boron mass 10 ; neutron,
mass 1 . The reaction can be symbolized
7
Li +4 He 10 B 1 n  2.8 MeV .
Let’s consider only the outgoing particles, boron and neutrons that are moving along the
same line as the initial helium nucleus, (call this the x -direction).
a) Draw momentum diagrams for the initial and final states in a reference frame in
which the target lithium nucleus is initially at rest.
b) What is the center of mass velocity of the system of lithium and helium? Does
this velocity change due to the collision?
c) Draw momentum diagrams for the initial and final states in a reference frame
moving with the velocity of center of mass.
d) What are the velocities of the helium nucleus and the lithium in the center of mass
reference frame? What is the momentum of the system in the center of mass
reference frame?
e) The minimum initial kinetic energy necessary for the reaction to take place is
called the threshold energy, E0,threshold . Draw a momentum diagram for the initial
and final states in the center of mass reference frame when the initial kinetic
energy of the incident particles is at the threshold energy (keep in mind the
lithium target is also moving in the center of mass reference frame). Hint: What
are the velocities of the outgoing particles at threshold?
f) Write down equations for conservation of energy and momentum in the center of
mass frame when the initial kinetic energy of the incident particles is at the
threshold energy. What is the threshold energy in the center of mass reference
frame?
g) Draw momentum diagrams for the initial and final states in the center of mass
reference frame when the initial kinetic energy of the incident particles lies in the
range E0,threshold  E0  E0,threshold  0.27 MeV . How many possible final states are
there? (Hint: Which way is the neutron moving?)
h) Draw momentum diagrams for the initial and final states in which the target is
initially at rest for the case of part g). How many possible final states are there?
(Hint: Which way is the neutron moving?)
i) What is the initial kinetic energy of the helium in this reference frame at
threshold? How many possible final states are there? (Hint: Which way is the
neutron moving?)