Download Impulse-Momentum Equation for Particles

Impulse-Momentum Equation for Particles
Linear, not Angular, Momentum: In this section, we deal with
linear momentum (mv) of particles only. Another section of
your book talks about linear (mvG) and angular (IGω)
momentum of rigid bodies.
An Integrated Form of F=ma: The impulse-momentum (I-M)
equation is a reformulation—an integrated form, like the workenergy equation—of the equation of motion, F=ma.
Particle Impulse-Momentum Equation
Important! This is a Vector Equation!
mv1 + Fdt = mv2
Initial
Linear Momentum
= mv1
Final
Linear Momentum
“Impulse”
= Fdt
= mv2
Derivation of the Impulse-Momentum Equation
Derivation of the Impulse-Momentum Equation
Typical Eqn of Motion:
Sub in a = dv :
dt
If mass is changing:
(e.g. for a rocket...)
(Newton wrote it this way...)
Separate variables:
Set up integrals:
F = ma
F = m dv
dt
d
F = dt
(mv)
Net Force =
time rate of
change of
momentum
F dt = m dv
t2
v2
F dt = m dv
t1
v1
t2
Integrate:
Usual form:
F dt = mv2 - mv1
t1
mv1 + Fdt = mv2
Particle Impulse-Momentum Equation
“Impulse” vs.
“Impulsive Force”
mv1 + Fdt = mv2
Initial
Momentum
Impulse
= Area under Force-time curve
= Any force acting over
any time....
e.g. 100 lb for two weeks....
Final
Momentum
“Impulse”
= Fdt
Impulse = Fdt
e.g. 1000 lb acting
in a 0.01 sec impact
Impulsive Force:
A relatively large force
Impulsive
which acts over a very short
Force
period of time, like in an impact,
e.g. bat-on-ball, soccer kick,
or 10 N acting
hammer-on-nail, etc.
for 1 second...
F
Non-Impulsive
Force
F
1000 lb
t
0.01 sec
10 N
1 sec
t
Area under
F-t curve
= 10 lb-sec
= Impulse
Area under
F-t curve
= 10 N-sec
= Impulse
Applications of the Impulse-Momentum Equation
For any problems involving F, v, t: The impulse momentum
equation may be used for any problems involving the variables
force F, velocity v, and time t. The IM equation is not directly
helpful for determining acceleration, a, or displacement, s.
Helpful for impulsive forces: The IM equation is most helpful
for problems involving impulsive forces. Impulsive forces are
relatively large forces that act over relatively short periods of
time, for example during impact. If one knows the velocities,
and hence momenta, of a particle before and after the action of
an impulse, then one can easily determine the impulse. If the
time of impulse is known, then one can calculate the average
force Favg that acts during the impulse.
For problems involving graph of F vs. t: Some problems give a
graph of Force vs. time. The area under this curve is impulse.
Important: You may need to find tstart for motion!
Various Forms of the Impulse-Momentum Equation
General Form:
Various Forms of the
I-M Equation
mv1 + Fdt = mv2
If force F is constant:
Force During Impact:
Actual Force
Often
Modeled as
mv1 + FAVG t = mv2
Know vectors v1 and v2:
F
FAVG
0
t
t
Impulse = Area under curve...
= F dt
= FAVG t
F dt = mv2 - mv1
= m( v2 - v1 )
If force F is constant:
FAVG t = mv2 - mv1
= m( v2 - v1 )
Impulse
produces
change in
momentum
I-M Equation Compared to the Work-Energy Equation
Comparison
Both derived
from F=ma
Impulse-Momentum Eqn
First write:
Integrate:
F=m
Equation
mv1 +
Work:
dv
dt
v2
F dt =
m dv
Integrate:
v1
1 mv2 +
1
2
Fdt = mv2
or
Equation
Explained
Vector or Scalar?
Applications
Impulse
F dt produces
a change in momentum,
dU = F ds = m ads
F = ma
Eqn of Motion:
Kinematics: a ds = v dv
t2
t1
Work-Energy Eqn
mv
A vector equation!
1 mv2 +
1
2
Work
2
dU =
1
v2
mv dv
v1
U1-2 = 1 mv22
2
Fds =
U1-2 or
a change in KE,
1
mv2
2
2
Fds effects
1
mv2
2
A scalar equation!
Problems involving v, F, t.
Problems involving v, F, s.
Not for problems needing
accel, a, or displacement, s.
Not for problems needing
accel, a, or time, t.