Momentum and Impulse

1
Momentum and Impulse

• Newtons original quantity of motion
• a conserved quantity
• a vector

Today

• Newtons Second Law in another form
• momentum and impulse

Serway Jewett 9.1 9.3
2
Definition The linear momentum p of a particle
is its mass times its velocity
p ? mv
Momentum is a vector, since velocity is a
vector. Units kg m/s (no special name).
(We say linear momentum to distinguish it from
angular momentum, a different physical quantity.)
3
The total momentum of a system of particles is
the vector sum of the momenta of the individual
particles
ptotal p1 p2 ... m1v1 m2v2 ...
Since we are adding vectors, we can break this up
into components so that
px,Tot p1x p2x .Etc.
4
Newtons Second Law
If mass is constant, then the rate of change of
(mv) is equal to m times the rate of change of v.
We can rewrite Newtons Second Law
or
net external force rate of change of momentum
This is how Newton wrote the Second Law. It
remains true in cases where the mass is not
constant.
5

• Relation between kinetic energy K and momentum
  p

(practice problem)
6
Example
Rain is falling vertically into an open railroad
car which moves along a horizontal track at a
constant speed. The engine must exert an extra
force on the car as the water collects in it (the
water is initially stationary, and must be
brought up to the speed of the train).
Calculate this extra force if
v 20 m/s
The water collects in the car at the rate of 6 kg
per minute
7
Solution
Plan The momentum of the car increases as it
gains mass (water). Use Newtons second law to
find F.
v is constant, and dm/dt is 6 kg/min or 0.1 kg/s
(change to SI units!), so F (0.1 kg/s) (20
m/s) 2.0 N
F and p are vectors we get the horizontal force
from the rate of increase of the horizontal
component of momentum.
8
Impulse
, or dp F dt
Newton 2
For a constant force, Dp F Dt . The vector
quantity F Dt is called the Impulse
I F?t ?p
(change in p) (total impulse from external
forces)
(Newtons Second Law again)
(Extra) In general (force not constant), we
integrate
Recall, the integral gives the area under a curve
9
Impulse is the area under the curve. The average
force is the constant force which would give the
same impulse.
Compare with work W F Dx so the work-energy
theorem (derived from Newton 2) is DK F Dx.
10
Example

• A golf ball is launched with a velocity of 44
  m/s. The ball has a mass of 50g. Determine the
  average force on the ball during the collision
  with the club, if the collision lasted 0.01 s.

11
Quiz

• A rubber ball and a snowball, each 100 grams, are
  thrown at a school bus window at identical
  speeds. The snowball sticks, the rubber ball
  bounces off. Which one transfers the larger
  impulse to the window?
• The rubber ball
• The snowball
• They both transfer the same amount

12
Example

• Figure below shows an approximate plot of
  force magnitude F versus time t during the
  collision of a 58 g ball with a wall. The initial
  velocity of the ball is 34 m/s perpendicular to
  the wall. The ball rebounds directly back with
  approximately the same speed, also perpendicular
  to the wall. What is Fmax, the maximum magnitude
  of the force on the ball from the wall during the
  collision?

13
Newton 3 and Momentum Conservation
Two particles interact
Dp1 F12 Dt Dp2 F21 Dt
Newtons 3rd Law F21 -F12
The momentum changes are equal and opposite the
total momentum p p1 p20
doesnt change.
The fine print Only internal forces act.
External forces would transfer momentum into or
out of the system.
14

• Since momentum is a vector, we can also
  express it in terms of the components. These
  are independently conserved
  pixpfx piypfy pizpfz

15
Application

• A subatomic particle may decay into two different
  particles. If the momentum before is zero, it
  must also be zero after, so

pi0
pf0
Given the mass and velocity of one of the
particles, if we measure the velocity of the
other one we can figure out its momentum and
hence the mass.
16
Example the fire extinguisher
How can we calculate the thrust on a fire
extinguisher when gas of mass Dm is ejected out
of the nozzle in time Dt ?
Plan Calculate the change in momentum of the
gas this is equal to the impulse the gas gets
from the extinguisher nozzle.
gas
thrust
From Newtons Third Law, the nozzle gets an equal
but opposite impulse from the gas. Since impulse
is thrust times Dt, divide by Dt to get the
average thrust (force).
Result thrust (speed of gas relative to
nozzle) times (mass of gas ejected per unit
time)
17
Summary

• Definitions momentum, impulse
• Newtons Second Law, in terms of momentum and
  impulse
• conservation of momentum