Momentum and Impulse
Linear Momentum (p)
defined as the product of an object's ÿlÿ' and 6/ÿ/ÿ cÿ"lÿ /ÿ'
I
I
p = my
I
I
momentum = mass x velocity
vector quantity (same direction as velocity)
units are ÿ ÿiS or /ÿ/ÿ
Examples:
a) different objects moving at same velocity small football player vs. a large football player
small car vs. a large truck
b) two objects with the same mass moving at different velocities two football players with the same mass traveling at different velocities
identlcal vehicleS traveling at different Velocities
Problem:
What is the momentum of a 75 kg football player moving at 8.0 m/s?
F
lImpulse (J) = Ap = mvf - mvo = FAt
l
" Both ÿ', ?ÿe_ and "ÿ'ÿ'ÿ are important in changing momentum.
Examples:
a) Hitting a concrete wall or a haystack while in a car
b) Dropping a dish on tile or plush carpet
c) Bungee jumping
Problems
1. A 1,000 kg car moving at 20 m/s slams into a building and comes to a halt. What impulse
acts on the car?
2. A car with a mass of 1000 kg moves at 20 m/s. What braking force is needed to bring the
car to a halt in 10 s?
j ÿ ÿ .pÿ -.ÿ ÿ.,ÿ.
Io
26
Summary,
Momentum is a vector quantity defined as the product of an object's mass and velocity. When two
objects interact (collide) with each other, momentum is conserved. The impulse of a force is equal to the
change in momentum of a body. This is also equal to the magnitude of the force multiplied by the length
of time the force is applied.
Vocabulary
impulse, conservation of momentum, elastic collision, momentum
Essential Questions
• What two factors affect momentum?
• How do you apply the law of conservation of momentum to determine unknown quantities in
collisions?
• In terms of impulse and momentum, why are air bags in automobiles a good idea?
Equations
Momentum = (mass) (,ÿelocity) .......
p =mv
Impulse = (force) (change in time) = (mass) (change in velocity)
J = FAt =mAv
Law of Conservation of Momentum
mÿvi, + m2v2, = mlvl, + mÿv2r
Practice Problems
1. During a classroom activity a suction-cup dart with a mass ofm was launched at a stationary
cart that had a mass of 5m. Four students observed the event, and their descriptions are shown in
the table. Which student best described the momentum, kinetic energy, and velocity of the
system before and after the collision?
After
Before
Students' Descriptions
::-sÿdÿntj:l:!:::i::, :;ÿ .... m. :!:-I:::i::ÿn tÿ£:ÿ, ergo:L ÿi-iAfÿf,ÿoliNOn, cdÿpa[ed:ÿlÿ :::
1
Yes
No
2
No
Yes
Less
3
Yes
Yes
Greater
4
No
No
Greater
Less
2. What is the impulse on a 45,000 kg airplane when it changes its velocity from 242 m/s to 258 m/n°
27
.
The first diagram shows two gliders, X and Y, on a nearly frictionless air track. A spring is
compressed between the two gliders. The gliders are tied together tightly by a piece of string.
The second diagram shows what happens to the gliders when the string is cut.
Before String iÿ; Cut
After String Is- Cut
3 n'¢ÿ ÿtÿ
ÿ 6 rn/is
................................ oo. .............................. ................
N.,
Which of the following quantities is the same for both gliders while the spring is pushing them
apart?
A. Magnitude of acceleration
B. Kinetic energy
....
C. Mass
D. Magnitude of momentum
....
p
4. A 6.0 x 10.3 kg insect is flying horizontally with a momentum of 4.8 x 10-2 kg m!s headwind.
What is its resulting/vÿrelative to the ground?
p
v_.l
5. A 0ÿm'flasÿight is constructed to withstand a specific foH
"
eÿ ÿ rce of impact that protects the bulb. A
standard 0.1 s is used as time-of-impact in calculaÿng a flashlight's specifications. What is the
n thekg0.4
flashlight
if it falls
and hits afloor
concrete 7.0y_ÿsv
at
2.5,,__meters
stoppinÿin 0.1ÿ7tp___ -ÿ seconds?t,qlt,ÿ- Mg' "ÿ kc'.''-ÿ'ÿ ÿ_ÿ,)_ ?
• ÿ-4-
. ÿ'qÿ:ÿfl ---,2, ÿ N
-ÿZ'. -
6. What impulse must be applied to a 25 kg cart to cause a velocity change of 12 m/s?
7. A ball is hit with a bat. A student determines that the momentum of the ball is 1.0 kg m/s. What
is tÿof the ball if it has a velocity of 2.0 m/s?
v/
u
P
8. A train car with a ;mass of 3.00 x l0ÿ kg traveling north at 1.5 m/s collides and couples with a
3.2 x 10ÿ kg train car going south at 0.8 m/s. What is the velocity of the coupled cars after the
.
x
9. TRUE or FALSE? A student on roller skates throws a basketball forward. The product of the
mass times the velocity of the student is equal in magnitude but opposite in direction to the
product of the mass times the velocity of the basketball.
28
Momentum and Impulse
1. Which has a greater mass, a heavy truck at rest or a rolling skateboard?
s" k,otÿ 6 oÿ,,X'
Which has greater momentum? ÿeÿ.¢ÿ ÿ-Pÿe,ÿ ÿ ÿ,ÿe ÿ
2. What is the momentum of a 50 kg carton that slide at 4.0 m/s across an icy surface?
(2.0 x 102 kg.m/s)"ÿ or ÿ ÿ;,ÿ_
3. The sliding carton in Problem ÿsÿds onto a rough surface and stops in 3.0 s. Calculate the
M ÿ ÿ'b ÿÿ---f°rce
of frictionÿ it, jencounters.ÿ(-67,;
ÿPÿ ÿ/ÿ-ÿ N)r_ÿ**drawÿ,
aÿpicture!!**%c_vÿv,./! ÿ.x9__ÿ5__ÿ°ÿ ÿ q)'
,,:.,./, /
....
......
..... /
........
4. If an average force of 10. N is exerted on a 2.0 kg cart for 2.5 s, calculate its final speed if
the cart is initially at rest. (13 m/s) ,
/ Iÿ0ÿ 2,ÿ 4 ÿ ÿ oÿ
/s
Momentum and Impulse Practice .... practice...practice.
Four steps still, show your work and units.
1. An ostrich with a mass of 146 kg is running with a velocity of 17 m/s. What was the ostrich momentum?
G/U
Equation
Sub
Solve
dÿl7
2,
A 21 kg child is riding a 5.9 kg bike with a velocity of 4.5 m/s. What was the momentum of both the bike and its rider?
G/U
Equation
Sub
Solve
,,¢ ¢ ÿ ÿ
P., ?
3°
What velocity must the ostrich have (from # 1 above) if she wanted to achieve a momentum of 3000 kgm/s?
29
,
A 2000 kg car moving at 15 m/s slams onto his brakes and comes to a complete stop. What impulse acts on the car?
ao
,
.ove
.
A 200 kg ostrich runs down the hall at Seven Lakes High School with a velocity of 10 m/s, Seven Lakes Physics
Teachers, because they are brave and know physics, stopped the ostrich in 7 seconds. How much force was applied?
G/O
Equationÿ = ÿ#
Sub
I Solve
/ÿ
V:ÿ 1oÿ, -r=
6, A 0.50 kg football is thrown with a velocity of 15 m/s. A stationary fan catches the ball and brings it to rest in 0.020
seconds, What is the force exerted on the fan?
G/Uvÿ..O
]Equation ÿ ÿ A,ÿ.____ -
,,,,ÿ ÿ.ÿ
j ÿ ÿf-- F,ÿ*
oÿ
ÿ__€ÿ
....
Sub
..........
(-?
,
If an average force of 20 N was exerted on a 4 kg cart for 5 seconds, what would be its final velocity if the cart was
already going 2 m/s?
G/U
Equation
Solve
rr.- 2.ÿv
o¢--tÿj,, aÿv; . rzÿ
too ar ÿ
Io 'ÿ
'ÿ- 4- ÿ u,'
Ve -- "ÿ
30
Conservation of Momentum
When two objects interact (collide) with each other, momentum is
Ap = 0 in a closed system.
Formula:
Types of collisions:
1. Elastic
a.
ÿoÿ. ÿ
is conserved
b. ÿa^ÿ4;ÿ ÿ_.ÿrÿV is conserved
c. ÿaÿ.(ÿ
2. Inelastic
a.
........
occurs, usuallyon ,ÿ'ÿ,ÿle,,ÿl
/
ÿ ÿ ÿ
is conserved
b. Kis yÿo÷ conserved
c. if objects .g¢,'Jÿ
...........
together, then
Problems:
1. mA = 2.0 kg, VA = +3.0 m/s and mB= 4.0 kg, vB = 0 find the velocity after the collision if both
massesare together after the collision. (va = vb after the collision) 7"/ÿ ;> iÿ elÿsÿ ÿ hÿ .,ÿ
2o mÿ = 5.0 g, vÿ = +10 m/s and mn = 10 g, vn = 0 after the cÿ1.0 m/s, find vnÿ. Is K Iÿ
conserved in i or 2?
3. A loaded rÿi ay ca
rolling
en it collides and
couples with an empty freight car of mass 3000 kg, rolling to the left on the same track at
3.0 m/s. What is the speed and direction of the pair after the collis'-ÿon (they arÿ
together)? ÿ1ÿ IJÿl; ÷ MÿVÿI -- (/NIÿ 4 Nÿ.) Vÿ
4. A 100. g ball moving at a constant velocity ot2OO-cÿ/s strikes a 400. g ball that is at rest.
After the collision, the first ball rebounds straight back at 120 cm/s. Calculate the final
velocity of the second ball. (80.0 cm/s [forward]) DRAW A PICTURE ÿeÿ ÿr
,ÿ %ÿ; • ÿtd ÿ; -: plÿ vÿ_4.ÿ elÿ veÿ
ÿ 0 ÿ ÿ 7
o)o+
, ÿ_ 4 o,rz-
.= t/g€
4o ÿ,ÿ r,'ÿ/,.t (
3ÿ
Class Work: Conservation of Momentum
,
A 75 kg girl running at 3.0 m/s jumps onto a sled that has a mass of 10. kg and that is already
moving in thee girl, at 2.0 m/s. What will be the final velocity of the girl
and the sled, assuming that the sle-'ÿ on level snow and that there is no friction? (2.9 m/s
[forward])
/ÿ1 ÿ/ÿ ÷ ÿiÿ ÿ/6!i ÿ- ÿ/ÿ 4 ÿlÿ) ÿ/ÿt
k_
,
A 1.5-kg wooden trolley on wheels is stationary on a horizontal, frictionless track. What will be
the final velocity of the trolley if a bullet of mass 21 g is fired into it with a horizontal velocity
of 300 m/s along the direction of the track? The bullet remains embedded in the trolley. (4.1
m/s [forward])
.
/ÿ ÿ/ÿ,i ÷ ÿB V/ÿ. ÿ- ('M ÿ • ÿ6) ÿ/p
A 24 g bullet is fired horizontally, embedding itself in a 10. kg block initially at rest on a
horizontal ice surface. The block slides along the ice, coming to rest in 2.0 s at a distance of
60.0 cm from its original position. Assuming that the frictional force stoppinÿ the block was
constant, calculate the velocity of the bullet. (250 m/s)
"-4
< . J€ 4- +
2,
.
g