Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Lecture 15
Chapter 11
Linear MomentumImpulse
Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI
Physics I
That’s what I do for taking me back to vectors
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Today we are going to discuss:
Chapter 11: Newton’s 2nd law (more general form): Section 11.1 Linear Momentum: Section 11.1 Impulse: Section 11.1
IN IN THIS CHAPTER, you will learn to use the concepts of impulse and linear momentum.
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Linear MomentumCh.11
Now we know how to use the energy approach to solve problems, so let’s go back to our old good Newton’s 2nd law, F=ma.
F=ma worked great for point objects, but it fails to explain systems such as rockets, where mass changes with motion
(Rockets accelerate by ejecting mass backward).
So, we need to modify (make it more general) our N. 2nd law somehow.
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Let’s rewrite N. 2nd law in terms of linear momentum
F ma ddt (m
v) m dv
dt d
pdt
The rate of change of momentum is equal to the net force
So, a force is required to change momentum of an object.
That is what actually Newton wrote.It is more general than our
earlier version F=madtpdF
v
m
Let’s pick an object and start describing it using our “old” N. 2nd law.
We got a new structure mv, so let’s give it a nice name and symbol
Linear momentum is defined as the product of an object’s mass and velocity:
vmp
So, finally, the Newton’s 2nd law will look this way:
Units of momentum: smkg
Momentum is a VECTOR!
ConcepTest Two Boxes/Momentum
A) –20 kg m/s
B) –10 kg m/s
C) 0 kg m/s
D) 30 kg m/s
The cart’s change of linear momentum Δpx is
Δpx = 10 kg m/s – (–20 kg m/s)
Negative initial momentum because motion is to the left and vx < 0.
x
x
initial
final
Final momentum
Initial momentum
∆
10 1 10
= 30 kg m/s
2
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
ImpulseCh.11
Now we have modified Newton’s 2nd law, let’s apply it to get something, i.e. impulse
How?
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Consider “Ball and Wall Collision”
The particle leaves with final velocity in the +x-direction.
(A large force exerted for a small interval of time is called an impulsive force)
(Before) The figure shows a particle with initial velocity in the −x-direction
The particle experiences an impulsive force of short duration Δt.
Now, let’s describe the collision using our new modified N. 2nd law (next slide)
F d
pdt
Instant of maximumcompression
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Ball and Wall Collision. Impulse
F d
pdt
Impulse= change in momentum
JdtFf
i
t
t
F dt
ti
t f dppipf if pp
Define Impulse as:
pddtF
From Newton’s 2nd law:
So, Impulse= area under F-vs-t curve
J
pJ
Integrate it:
During the collision, objects are deformed because of the large forces involved . How to relate those forces with a change in momentum?
Force exerted on the ball
Befo
re c
ollis
ion
Afte
r col
lisio
n
fp
ip
Recall: Integral Graphical meaning is an area
p
Work-KE Principle
ImpulsepJ
f
i
t
t
dtFJ
Momentum Principle
Similarity
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
The force exerted by a tennis racket on the ball (mass 56 g) during a serve ( ) can be approximated by the F vs time plot below.What is the impulse on the ball? What is the speed of the serve?
m/s 71kg 056.0sN 4
fv
Forc
e (k
N)
Time (ms) 10
2
Area under force-time curve is an impulse:
JdtFf
i
t
t
pf pi
AreaJ
ff mvpJ
0
mJv f
0iv
Tennis Ball/Racket CollisionExample
1 2 3 7
)2
22(2 mskN sN 4
5A
B
C
)(2 ABCArea
pJ
Momentum Principle
ConcepTest Kicking BallA) 0.5 m/s left
B) At rest
C) 0.5 m/s right
D) 1.0 m/s right
E) 2.0 m/s right
x
A 2.0 kg object moving to the right with speed 0.50 m/s experiences the force shown. What are the object’s speed and direction after the force ends?
x
vi=0.5m/s
vf -?
Δpx = Jxpf - pi = Jxpf = pi + Jx
∆
∆
. .
/
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Impulse/Average force
J
The exact variation of F with time is very often not known (too complicated).
How to get at least something?
J
ptFavg
ptFdtF avgt
t
f
i
Let’s keep the same impulse JAnd the same time interval Δt
But instead of a real forcewe will use an average force
It is easier to find an average force.
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Having a certain ∆p, a cat by bending its lags tries to increase ∆t (impact time), so that an impact force would be reduced. (intuitive knowledge of Physics )
ptFavg
How to avoid broken legs for a cat?Example
Initial linear momentum
Final linear momentum0
A
Since the cat falls from a certain height, p(initial)=Δp is given and the cat cannot do anything about that during the collision.
F, that is what can break cat’s bones and the cat “feels” that and tries to reduce F as much as it can.
By bending legs and increasing an impact time, Δt.
Collision with a floorFavg
ConcepTest Two Boxes/Momentum
F F light heavy
We know:
In this case F and t are the same for both boxes!Both boxes will have the same final momentum.
A) the heavier one
B) the lighter one
C) both the same
Two boxes, one heavier than the other,are initially at rest on a horizontalfrictionless surface. The same constantforce F acts on each one for exactly1sec. Which box has more linearmomentum after the force acts ?
tpp if
tFp avf 0iptpFav
(In the previous situation)
Which box has the larger
velocity after the force acts?
A) the heavier one
B) the lighter one
C) both the same
ConcepTest Two Boxes/velocity
lfh mvpMv
hl vvthenmMSince ,
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
The speed of a fastball is about 40 m/s, and the speed of the ball coming off of player’s bat for a home run is about 54 m/s. The ball (0.145kg) is in contact with the bat for 1ms. What is the average Force exerted by the player?
NFaverage 300,136
tpp
F ifaverage
ssmkgFaverage 001.0
/]4054)[145.0(
tvvm if
))((
in the direction of xorv f
x
Pay attention to directions!!!!!!!!
Faverage
p
t
Average Force on a baseballExample
ptFavg
Department of Physics and Applied PhysicsPHYS.1410 Lecture 15 Danylov
Thank you