Download Physics 1422 - Introduction

Physics 203 – College Physics I
Department of Physics – The Citadel
Physics 203
College Physics I
Fall 2012
S. A. Yost
Chapter 7 Part 2
Momentum and Collisions
Physics 203 – College Physics I
Department of Physics – The Citadel
Announcements
Read Ch. 7, but you can skip sections 7.7 and 7.9.
A problem set HW07A was due today.
A problem set HW07B on sections 4 – 6 (collisions)
is due Thursday.
Next class: Rotational Motion, Ch. 10.
Read sec. 1 – 5 for next time.
Homework set HW08 on sections 1 – 6 will be due
next Thursday. Sections 7 – 8 will be combined
with chapter 9 in a later set. Skip section 9.
Physics 203 – College Physics I
Department of Physics – The Citadel
Exam 2
The average was 36%, high 67.5%.
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Physics 203 – College Physics I
Department of Physics – The Citadel
Momentum and Impulse
Momentum is defined to be
→
→
p=mv
The change in momentum is the impulse.
→
→
→
D p = Favg t = J
This is a consequence of Newton’s 2nd Law.
In an isolated system of objects, the total momentum
of the objects is always conserved.
This is a consequence of Newton’s 3rd Law.
Physics 203 – College Physics I
Department of Physics – The Citadel
1D Collisions
In a one dimensional collision, two masses m1
and m2 approach with velocities v1 and v2,
and then collide. Their velocities v1’ and v2’
are measured after the collision.
m1
m2
m1 v1 + m2 v2 = m1 v1’ + m2 v2’
Dp1 + Dp2 = 0
Physics 203 – College Physics I
Department of Physics – The Citadel
Elasticity of a Collision
Not all collisions are equally “bouncy”.
This is quantified by elasticity.
In a perfectly elastic collision, the relative speeds of
the colliding objects is the same, though the
direction may be different.
Physics 203 – College Physics I
Department of Physics – The Citadel
Inelastic Collision
One possibility is that the red ball can come in with
velocity v, stick to the blue ball, and both can travel
off with final velocity v’.
v’
v
m
m
Physics 203 – College Physics I
Department of Physics – The Citadel
Inelastic Collision
This case is called an inelastic collision.
In this case, the relative velocity of the
masses is zero after the collision, since
the balls stick together.
m1 m2
v’
Physics 203 – College Physics I
Department of Physics – The Citadel
Example
For example, a railroad car of mass M1 = 1200 kg
could travel at v = 2.5 m/s and strike a second
railroad car of mass M2 = 2800 kg. If the cars
collide and hitch together, how fast do they
move together?
2.5 m/s
1200 kg
2800 kg
Physics 203 – College Physics I
Department of Physics – The Citadel
Example
Initial momentum:
pi = M1 v = 1200 kg x 2.5 m/s = 3000 kg m/s.
Final momentum: pf = (M1 + M2) v’
v’ = 3000 kg m/s / 4000 kg = 0.75 m/s.
2.5 m/s
1200 kg
2800 kg
Physics 203 – College Physics I
Department of Physics – The Citadel
Energy in a Collision
How much energy is converted to internal energy
(vibrations, heat, etc.) in this collision?
Ei = ½ M1 v2 = ½ (1200 kg)(2.5 m/s)2 = 3750 J.
Ef = ½ (M1+M2)v’2 = ½ (4000 kg)(0.75 m/s)2
= 1125 J.
Ei – Ef = 2625 J.
Physics 203 – College Physics I
Department of Physics – The Citadel
Question
Which of the inelastic collision(s) shown
bring(s) the car at left to a halt? Choose D if
all of them, E if none of them.
A
B
C
Physics 203 – College Physics I
Department of Physics – The Citadel
Question
In which collision does the car on the left
experience the greatest force? Choose D if
they are all equal, E if more information is
needed.
A
B
C
Physics 203 – College Physics I
Department of Physics – The Citadel
Question
In collision C, which car experiences the
greater force?
A = left,
A
B
C
B = right,
C = equal.
Physics 203 – College Physics I
Department of Physics – The Citadel
Question
Which of the labeled cars has the greatest
change in energy, comparing all of the cases
together? Pick A, B, C, D, or E. (One of
these is correct.)
A
B
D
C
E
Physics 203 – College Physics I
Department of Physics – The Citadel
Elastic Collision
An elastic collision conserves energy.
In one dimension, a collision is elastic if the
relative speeds are the same,
but notice the
sign reversal
v 2’ – v 1’ = v 1 – v 2.
m1
m2
v1
m2
m1
v2
v1’
v2’
Physics 203 – College Physics I
Department of Physics – The Citadel
Bat and Ball Example
v’ = 56 m/s
v = - 35 m/s
M = 850 g
m = 145 g
A bat is swung at a ball pitched at 35 m/s,
hitting the ball back on a line-drive at 56 m/s.
What was the speed of the bat before and after
the collision if the bat has mass M = 850 g?
Assume an elastic collision.
Physics 203 – College Physics I
Department of Physics – The Citadel
Bat and Ball Example
v2’ = 56 m/s
v2 = - 35 m/s
m1 = 850 g
m 2= 145 g
Let v1, v1’ be the speed of the bat before and
after the collision.
Momentum conservation :
m1v1 + m2v2 = m1v1’ + m2v2’
Elastic collision: v1 – v2 = v2’ – v1’
Physics 203 – College Physics I
Department of Physics – The Citadel
Bat and Ball Example
m 1 v 1 + m2 v 2 = m 1 v 1 ’ + m 2 v 2 ’
m1v1 – m1v2 = m1 v2’ – m1 v1’
2m1v1 + (m2 – m1) v = (m1 + m2) v’
2m1v1 = (m1 + m2) v’ + (m1 – m2) v
1.7 v1 = 0.995(56 m/s) + 0.705(–35 m/s)
= 31.0 m/s
v1 = 18 m/s.
v1’ = v2’ + v2 – v1 = 3 m/s.
Physics 203 – College Physics I
Department of Physics – The Citadel
Force Due to a Stream of Mass
This mini-UZI fires 16 rounds per second,
with m = 8g each, into a wall at velocity
v0 = 360 m/s. What is the average force
on the wall?
Think about the impulse
resulting in the change of
momentum when the
bullets are stopped by
the wall.
Physics 203 – College Physics I
Department of Physics – The Citadel
Force Due to a Stream of Bullets
The wall stops the bullets, applying a force to
the bullets of Fwb = dp/dt. In time t = 1/16 s, a
mass m = 8 g hits the wall.
The momentum lost is
Dp = -m v0.
The impulse is
Fwb = Dp/t
= -mv0/t = - 46 N
Fwb
Physics 203 – College Physics I
Department of Physics – The Citadel
Force Due to a Stream of Bullets
The stream of bullets could be used as a
propulsion mechanism for the gun… this is
exactly the mechanism of rocket propulsion.
The wall plays no role in this; the gun pushes
against the bullets.
F = 46 N
Physics 203 – College Physics I
Department of Physics – The Citadel
Question
Two people on roller blades throw a ball back
and forth. After a couple of throws, they are
(ignore friction)
A. standing where they were initially.
B. standing farther away from each other.
C. standing closer together.
D. moving away from each other.
E. moving toward each other.
Physics 203 – College Physics I
Department of Physics – The Citadel
Recoil
Suppose a box of mass
m = 10 kg is thrown out
of a small boat of mass
M = 100 kg with a
speed relative to the
boat of 15 m/s.
m
M
What is the speed of the boat after the
box is thrown?
Physics 203 – College Physics I
Department of Physics – The Citadel
Recoil
Momentum conservation
implies
MV + mv = 0.
We were given the speed
of the box relative to
the boat:
v0 = 15 m/s = v – V
v
m
V
M
Physics 203 – College Physics I
Department of Physics – The Citadel
Recoil
MV + mv = 0.
v
m
V
v = v0 + V.
M
MV + m(v0+ V) = 0.
Solving for V gives
V=
-m v0
-(10 kg) (15 m/s)
M+m
=
110 kg
= - 1.4 m/s
Physics 203 – College Physics I
Department of Physics – The Citadel
Center of Mass
The center of mass a set of objects is the average
position of their mass. For two objects in 1D:
x cm =
x 1 m1 + x 2 m2
m1 + m 2
x1
x2
CM
m2
m1
xcm
Physics 203 – College Physics I
Department of Physics – The Citadel
Center of Mass
In more dimensions you can use vectors to
locate the CM:
m1r1 + m2r2 + m3r3
rcm =
m1 + m2 + m3
It moves with velocity
vcm =
m1 v1 + m2v2 + m3v3
m1 + m2 + m3
m2
m1
r1
r2
CM
m2
r3
P = M vcm
Physics 203 – College Physics I
Department of Physics – The Citadel
Motion of the Center of Mass
If an external force F acts on an extended object or
collection of objects of mass M, the acceleration of the
CM is given by
F = Macm.
You can apply Newton’s 2nd Law as if it were a particle
located at the CM, as far as the collective motion is
concerned.
This says nothing about the relative motion, rotation,
etc., about the CM. That comes up in chapter 8.
Physics 203 – College Physics I
Department of Physics – The Citadel
Motion of Extended Objects
The motion of extended objects or collections of
particles is such that the CM obeys Newton’s 2nd
Law.
Physics 203 – College Physics I
Department of Physics – The Citadel
Motion of the Center of Mass
The CM of a wrench sliding on a frictionless table will move in
a straight line because there is no external force. In this
sense, the wrench may be though of as a particle located
at the CM.
cm motion
Physics 203 – College Physics I
Department of Physics – The Citadel
Motion of the Center of Mass
For example, if a hammer is thrown, its CM follows a
parabolic trajectory under the influence of gravity,
as a point object would.
Physics 203 – College Physics I
Department of Physics – The Citadel
Motion of the Center of Mass
For example, if a hammer is thrown, its CM follows a
parabolic trajectory under the influence of gravity,
as a point object would.