Physics I
Class 07
Momentum and
Impulse
07-1
Momentum of an Object
Definitions
We define momentum
for an object to be:

p  mv
Momentum is a vector. It is in the same direction as velocity.
SI units for momentum: kg m/s.
Later in the semester, we will talk about angular momentum.
Plain “momentum” is also known as linear or translational momentum.
07-2
Change of Momentum
Change of momentum is the difference between the final value
and the initial value.
Beware: We are subtracting vectors!
 



 p  p final  p initial  m v final  v initial 
final
initial
+3

+3
= 0 kg m/s
+3

-3
=
+6 kg m/s
07-3
Connecting Net Force and the
Derivative of Momentum
Newton’s Second
 Law:

 F  Fnet  m a
The time derivative of momentum:


dp d
d
v


 ( m v)  m
 ma
dt dt
dt
We can write Newton’s Second
Law in a new form:

 
dp
 F  Fnet  d t
07-4
The Fundamental Theorem of
Calculus
Recall from the first lecture (the second day of v
class):
Math Fact: Because velocity is the derivative
v0
of displacement, displacement is the area
(integral) under the graph of v versus t.
The Fundamental Theorem of Calculus:
If f is the derivative of g, then
t0
t
b
 f (t ) dt  g(b)  g(a )   g
a
07-5
Impulse and the ImpulseMomentum Theorem
Impulse is defined to be the time integral of force. SI units = N s.
Like force, it is a vector. (Net or total force is implied.)


J   F dt
Using the Fundamental Theorem of Calculus:



J   F dt   p
In Physics, this is known as the Impulse-Momentum Theorem.
07-6
Example Problem Using
Impulse-Momentum
An object of mass 0.5 kg is subjected to a
F
force in the +X direction that varies as shown 5N
in the graph from 0 to 7 seconds. Its initial X
velocity is zero. What is its final X velocity?
Doing this with F=ma would be hard. Doing 0
0
it with impulse-momentum is much easier.
7s t
J = area = 2+3+3.5+3+2+3+1.5 = 18 N s.
18 N s  J   p  p final  p initial
p final  p initial  p final  0  p final  18 kg m / s
v final  p final  m  36 m / s
07-7
One-Dimensional Impulse
and Average Force
If the problem is one-dimensional, we deal
in scalar components:
J   F dt   p
F
5N
Favg = 18/7
Average force is the constant force that
would produce the same impulse over a
0
specified time interval, t:
Favg  t  J   F dt   p
0
7s t
07-8
Class #7
Take-Away Concepts
1.
Momentum

 defined for an object:
2.
A new way to write Newton’s Second Law:
3.
Impulse defined:
4.
Impulse-Momentum Theorem:
5.
Average Force over time interval t (one dim.):
p  mv

 
dp
 F  Fnet  d t


J   F dt



J   F dt   p
Favg  t  J   F dt   p
07-9
Activity #7 Impulse and Momentum
Objectives of the Activity:
1.
2.
3.
4.
Perform a preliminary “thought” experiment to study the
relationships among average force, time interval,
impulse, and momentum change.
Learn how to calibrate sensors to take accurate data.
Take data in a real experiment to study the relationship
between impulse and momentum change.
Learn how to use data analysis features of LoggerPro.
07-10
Class #7 Optional Material
Special Theory of Relativity
Albert Einstein (1879–1955)
In Lecture 03, we mentioned that Albert Einstein
showed that Newton’s Second Law had to be
modified in order to account for the observed
behavior of electromagnetic waves (light) and the
interaction of electromagnetic fields with matter.
Einstein’s Two Postulates of Special Relativity:
1. The laws of physics are the same in all inertial frames.
2. The speed of light, c, is constant in all inertial frames.
07-11
What is an Inertial Frame?
An inertial frame (of reference) is a real or imaginary set of devices
for measuring position and time that are in motion together
according to Newton’s First Law; in other words, these devices are
not accelerating (or rotating).
Neglecting gravity (we’ll talk about that later in the semester), the
track and motion detector that we use in our activities, along with the
clock in your PC when you run LoggerPro, comprise an inertial
reference frame with a special name: the laboratory reference
frame (because this is the frame we use to make measurements).
If we were to set the same equipment up in the Ferris Wheel we
studied earlier, that would not be an inertial reference frame.
Instead, we call that an accelerated frame.
07-12
Where Did Einstein’s
Postulates Come From?
1. The laws of physics are the same in all inertial frames.
This idea goes back to Galileo. Imagine an experiment like our cart track
and hanging weight being performed in an airliner moving uniformly in one
direction at a constant speed and altitude. (Assume no turbulence and the
altitude is low enough so that the force of gravity is about the same as on
the ground.) Our measurements that we take with LoggerPro should be the
same as what we did in class if we set things up carefully.
2. The speed of light, c, is constant in all inertial frames.
Nobody wanted to believe this prior to the Michelson-Morley experiment in
1887. In one of the most famous null results in the history of science, they
showed by careful measurement that the speed of light is independent of the
relative motion of the source and detector. This counter-intuitive result was
not adequately explained until Einstein’s Special Theory of Relativity. Do a
web search and you will see that many people still refuse to believe it!
07-13
Einstein’s Correction to
Newton’s Second Law
Starting from the two postulates, Einstein showed that Newton’s
Second Law would be correct for all velocities if the definition of
momentum was modified:

 
dp
F

F

 net d t

p
m
v2
1 2
c

v
Note that the square root term in the denominator allows the
magnitude of the momentum to become arbitrarily large as the
speed approaches but never quite reaches the speed of light. At
normal speeds much smaller than c, the correction is negligible.
07-14