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AP PHYSICS C
Momentum
Background
A particle’s momentum p is defined as the product of its mass and velocity:
p=mv
Momentum is a vector quantity that may be thought of as a measurement of the effort needed to
bring a particle to rest.
Newton’s second law can be written in terms of the momentum of a particle. Differentiating the
previous equation, we obtain:
dp d(mv)
dv
=
= m = ma
dt
dt
dt
Then substituting the force Fnet for ma, we get
†
Fnet =
dp
dt
Thus the net force acting on a particle equals the time rate of change of the particle’s linear
†
momentum.
The total momentum P of a system of many particle is the sum of the momenta of the individual
particles:
P = Â m i vi = Â p i
i
†
i
The momentum sum equals the total mass M times the velocity of the center of mass:
P = Â mi vi = Mvcm
i
Differentiating with respect to time, we obtain:
†
dP
dv
= M cm = Macm
dt
dt
But according to Newton’s second law, Macm equals the net external force acting on the system:
†
ÂF = F
ext
i
net .ext
=
dP
dt
When the net external force acting on a system of particles is zero, the rate of change of the total
momentum is zero, and the total momentum of the system remains constant:
†
P = Â mi vi = Mvcm = cons tan t
i
The result is known as the law of conservation of momentum:
If the net external force on a system is zero, the total momentum of the
†
system remains constant.
Example 1:
During repair of the Hubble Space Telescope, an astronaut replaces two solar panels whose
frames are bent. Pushing the detached panels away into space, she is propelled in the opposite
direction. The astronaut’s mass is 60 kg and the panel’s mass is 80 kg. The astronaut is at rest
relative to her spaceship when she shoves away the panel and she shoves it at 0.3 m/sec relative
to the spaceship. What is her subsequent velocity relative to the space ship?
Example 2:
A runaway 14000 kg railroad car is rolling at 4 m/sec toward a switchyard. A sudden downpour
fills the open-topped car with 2000 kg of rainwater. After the rainstorm, how long does it take
the car to cover the 500 m distance to the switchyard? Assume that the rain comes straight down
and that slowing due to friction is negligible.
Example 3:
A 40 kg skateboarder on a 3 kg board is training with two 5 kg weights. Beginning from rest, she
throws the weights horizontally one at a time from her board. The velocity of each weight is 7
m/sec relative to her and the board after it is thrown. How fast is she propelled in the opposite
direction after throwing the second weight? Assume the board rolls without friction.
Example 4:
A thorium 227 nucleus decays into a radium-223 nucleus (mass 223 u) by emitting an alpha
particle (mass 4 u). The kinetic energy of the alpha particle is found to be 6.00 MeV. What time
is the kinetic energy of the recoiling radium nucleus?