1Chapter 9 Linear MomentumDescartes believed that God had
created the world like a perfect and never-changing clockwork
mechanism, and asserted that the total quantity of motion, which he
defined as mass x speed, would remain constant.In 1668, scientists
began to communicate their result and soon realized that the
quantity of motion was conserved if it was re-defined as mass x
velocity.Although Descartes analysis of specific examples was weak,
he had sown the seed of an extremely important idea in physics:
There is some physical quantity that does not change within an
isolated system. This is called a principle of conservation.2Linear
Momentum and Kinetic EnergyIn the meantime, Huygens and Wren
independently concluded the quantity mv2is conserved in collisions
between hard balls. Newton found the vector mv was always
conserved, but that the scalar quantity mv2was conserved only in
special case of collisions between hard spheres.Later, Newton
conducted carefully experiments on collision between all kinds of
substances, such as glass, wood, steel, and putty.3Force and
MomentumIn Newtons own version of his second law, he stated that
the motive force exerted on a particle is equal to the change in
its linear momentum. In 1752, Euler modified Newtons definition to
include the aspect of time explicitly. A modern statement of
Newtons second law is:The net force acting on a particle is equal
to the time rate of change of its linear momentum.dtdPF =49.2
Conservation of Linear Momentum2 2 1 1 2 2 1 1v v u u m m m m + =
+The initial and final velocities are related according to the
principle of the conservation of linear momentum:If the net
external force acting on a system is zero, the total linear
momentum is constant.In order to apply the conservation of linear
momentum, there must be no net external force acting on the
system.dtdextPF == = = constantthen , 0 IfiP P Fext5Type of
Collision2 2 1 1 2 2 1 1v v u u m m m m + = +Collisions may be
either elastic or inelastic. Linear momentum is conserved in both
cases. A perfectly elasticcollision is defined as one in which the
total kinetic energy of the particles is also
conserved.Super-elastic collision refers to the possibility that
total kinetic energy increases as a result of collision. This is
because the collision triggers a system to release extra potential
energy.21 221 122 221 121212121v v u u m m m m + = +elastic and
inelasticelastic6Example 9.1A 2000-kg Cadillac limousine moving
east at 10 m/scollides with a 1000-kg Honda Prelude moving west at
26 m/s. The collision is completely inelastic and takes place on an
icy (frictionless) patch of road. (a) Find their common velocity
after the collision. (b) what is the fractional loss inkinetic
energy?99 . 0 438000 / ) 438000 6000 ( / ) (J 438000 26 ) 1000
(2110 ) 2000 (21J 6000 ) 1000 2000 (21(b)m/s 2 3000 26 1000 - 10
2000 (a)2 22 = = = + == + == = initial initial finalinitialafter
finalafter afterE E EEv Ev vSol:7Example 9.2A 3.24-kg Winchester
Super X rifle, initially at rest, fires a 11.7-g bullet with a
muzzle speed of 800 m/s. (a) What is the recoil velocity of the
rifle? (b) What is the ratio of the kinetic energies of the bullet
and the rifle?% . / . /K Kv m Kv m Kv v v .b gg g gb b b33 0 3744 5
12 J 5 . 12 88 . 2 24 . 3 5 . 021 J 3744 800 0117 . 0 5 . 021(b)m/s
88 . 23240 ) 800 ( 7 11 onconservati momentum (a)2 22 2= == = == =
== = Sol:8Example 9.5In 1742, Benjamin Robins devised a simple, yet
ingenious, device, called a ballistic pendulum, for measuring the
speed of a bullet. Suppose that a bullet , of mass m=10 g and speed
u, is fired into a block of mass M= 2 kg suspended as in Fig. 9.8.
The bullet embeds in the block and raises it by a height h=5 cm.
(a) How can one determine u from H? (b) What is the thermal energy
generated?Sol:) ( onconservati Energy ) ( onconservati MomentumV,
Hu,V99.3 Elastic Collision in One Dimension2 2 1 1 2 2 1 1v m v m u
m u m + = +In a one-dimension elastic collision,the relative
velocity is unchanged in magnitude but is reversed in direction.21
221 122 221 121212121v m v m u m u m + = +momentumkinetic energy)
(1 2 1 2u u v v = elastic(9.10)10Elastic Collision in One
Dimension2 11 122 11 2 112) (m mu mvm mu m mv+=+=(i) Equal masses:
m1=m2=mvelocities exchange,1 2 2 1u v u v = =(ii) Unequal masses:
m1 m2 , Target at rest: u2=0119.4 ImpulseThe impulse I experienced
by a particle is defined as the change in its linear momentum:For
the same impulse, prolonging the interacting time will reduce the
average applied force.Examples: jump from a wall and catch a ball.
i fP P P I = =Impulse is a vector quantity with the same unit as
linear momentum (kgm/s). dt t tttt) (210= = = F F P I129.5
Comparison of Linear Momentum with Kinetic Energy(i) Conservation
of linear momentum is a general valid law, whereas the conservation
of kinetic energy is true only in the special case of elastic
collision.dx Kdtd= = , FPF(ii) Momentum is a vector, whereas the
kinetic energy is a scalar.(iii) Force, momentum, and kinetic
energy are all co-related concepts.139.6 Elastic Collision in Two
Dimension2 2 2 1 1 1 1 1cos cos v m v m u m Px+ =There are four
unknowns v1, v2, c 1, and c 2 in three equations.Often, either c 1
or c 2 is measured.2 2 2 1 1 1sin sin 0 v m v m Py =22 2 2121 1
2121 1 21v m v m u m K + =14Example 9.9A proton moving at speed u1=
5 km/s makes an elastic collision with another proton initially at
rest. Given that c1=37X, find v1, v2and c2. Sol:222122 2 12 2 15on
conservati Energy sin 6 . 0axis - y at cos 8 . 0 5axis - at xon
conservati Momentumv vv vv v+ = = + =15Exercises and ProblemsCh.9:
Ex.10, 12, 17, 30Prob. 4, 5, 18, 19
