9.2 A Model System to Represent a Dilute Gas
389
Exercise 9.3
Assume that a particle of mass m moves in the z direction and is subject to a constant gravitational
force given by F z −mg. If the initial position of the particle is z(0) 0 and its initial velocity
is v z (0) 0, solve its equation of motion, finding v z and z as functions of time.
Newton’s third law is called the law of action and reaction. It asserts that if one object
exerts a force on a second object, the second object exerts a force on the first that is
equal in magnitude and opposite in direction to the first force. If F 21 is the force exerted
on the second object and if F 12 is the force exerted on the first object, then
F 12 −F 21 (Newton’s third law)
(9.2-5)
In order to jump into the air from a floor, you push on the floor with your feet, and the
floor throws you into the air because of Newton’s third law.
Potential Energy
If the force on a particle depends only on its position, the force can be derived from a
potential energy. For a force in the z direction,
F z −
∂V
∂z
(9.2-6)
where the potential energy is denoted by V . This derivative is a partial derivative, which
means that x and y are treated as constants in the differentiation. Similar equations apply
for the x and y directions. If the force on a particle depends only on its position, the total
energy of the particle is the sum of its kinetic energy K and its potential energy V . If
the particle moves only in the z direction, V depends only on z:
E K + V
1
2
mv
2
z + V (z)
(9.2-7)
The law of conservation of energy asserts that the energy of a particle subject only
to forces from a potential energy is constant. We say that the energy is conserved and
that the energy is a constant of the motion.
Exercise 9.4
a. Show that the solution to Exercise 9.3 conforms to the conservation of energy. The gravitational potential energy V corresponding to F z −mg is
V mgz + constant
where the constant can take on any value without any physical effect.
b. Explain why the conservation of kinetic plus potential energy does not apply to the solution
of Example 9.1. Because the conservation of total energy is an accepted principle of science,
the decrease in kinetic energy in this example must correspond to an increase of some other
form of energy. What form is this?
389
Exercise 9.3
Assume that a particle of mass m moves in the z direction and is subject to a constant gravitational
force given by F z −mg. If the initial position of the particle is z(0) 0 and its initial velocity
is v z (0) 0, solve its equation of motion, finding v z and z as functions of time.
Newton’s third law is called the law of action and reaction. It asserts that if one object
exerts a force on a second object, the second object exerts a force on the first that is
equal in magnitude and opposite in direction to the first force. If F 21 is the force exerted
on the second object and if F 12 is the force exerted on the first object, then
F 12 −F 21 (Newton’s third law)
(9.2-5)
In order to jump into the air from a floor, you push on the floor with your feet, and the
floor throws you into the air because of Newton’s third law.
Potential Energy
If the force on a particle depends only on its position, the force can be derived from a
potential energy. For a force in the z direction,
F z −
∂V
∂z
(9.2-6)
where the potential energy is denoted by V . This derivative is a partial derivative, which
means that x and y are treated as constants in the differentiation. Similar equations apply
for the x and y directions. If the force on a particle depends only on its position, the total
energy of the particle is the sum of its kinetic energy K and its potential energy V . If
the particle moves only in the z direction, V depends only on z:
E K + V
1
2
mv
2
z + V (z)
(9.2-7)
The law of conservation of energy asserts that the energy of a particle subject only
to forces from a potential energy is constant. We say that the energy is conserved and
that the energy is a constant of the motion.
Exercise 9.4
a. Show that the solution to Exercise 9.3 conforms to the conservation of energy. The gravitational potential energy V corresponding to F z −mg is
V mgz + constant
where the constant can take on any value without any physical effect.
b. Explain why the conservation of kinetic plus potential energy does not apply to the solution
of Example 9.1. Because the conservation of total energy is an accepted principle of science,
the decrease in kinetic energy in this example must correspond to an increase of some other
form of energy. What form is this?