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**Unformatted text preview: **ation as functions of
time. Cause-and-effect relationships exist among these quantities: Velocity causes
position to change, and acceleration causes velocity to change. Because acceleration is the direct result of applied forces, any analysis of the dynamics of a particle
usually begins with an evaluation of the net force being exerted on the particle.
Up till now, we have used what is called the analytical method to investigate the
position, velocity, and acceleration of a moving particle. Let us review this method
brieﬂy before learning about a second way of approaching problems in dynamics.
(Because we conﬁne our discussion to one-dimensional motion in this section,
boldface notation will not be used for vector quantities.)
If a particle of mass m moves under the inﬂuence of a net force F, Newton’s
second law tells us that the acceleration of the particle is a
F/m. In general, we
apply the analytical method to a dynamics problem using the following procedure:
1.
2.
3.
4. Sum all the forces acting on the particle to get the net force F.
Use this net force to determine the acceleration from the relationship a
F/m.
Use this acceleration to determine the velocity from the relationship dv/dt a.
Use this velocity to determine the position from the relationship dx/dt v.
The following straightforward example illustrates this method. EXAMPLE 6.15 An Object Falling in a Vacuum — Analytical Method Consider a particle falling in a vacuum under the inﬂuence
of the force of gravity, as shown in Figure 6.18. Use the analytical method to ﬁnd the acceleration, velocity, and position of
the particle.
2 Solution The only force acting on the particle is the
downward force of gravity of magnitude Fg , which is also the
net force. Applying Newton’s second law, we set the net force
acting on the particle equal to the mass of the particle times The authors are most grateful to Colonel James Head of the U.S. Air Force Academy for preparing
this section. See the Student Tools CD-ROM for some assistance with numerical modeling. 170 CHAPTER 6 Circular Motion and Other Applications of Newton’s Laws its acceleration (taking upward to be the positive y direction):
Fg ma y In these expressions, yi and vyi represent the position and
speed of the particle at t i 0. mg g, which means the acceleration is constant. BeThus, a y
g, which may be incause dv y /dt a y, we see that dv y /dt
tegrated to yield
v y(t) v yi gt Then, because v y dy/dt, the position of the particle is obtained from another integration, which yields the well-known
result
y(t) yi v yi t 12
2 gt mg Figure 6.18 An object falling in vacuum under the inﬂuence of gravity. The analytical method is straightforward for many physical situations. In the
“real world,” however, complications often arise that make analytical solutions difﬁcult and perhaps beyond the mathematical abilities of most students taking introductory physics. For example, the net force acting on a particle may depend on
the p...

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