2.7 There are certain simple one-dimensional problems where the equation of motion (Newton's second law) can always be solved, or at least reduced to the problem of doing an integral. One of these (which we have met a couple of times in this chapter) is the motion of a one-dimensional particle subject to a force that depends only on the velocity v, that is, F = F(v). Write down Newton's second law and separate the variables by rewriting it as m dv/F(v) = dt. Now integrate both sides of this equation and show that "S" Vo dv' F (v') t=m Provided you can do the integral, this gives t as a function of v. You can then solve to give v as a function of t. Use this method to solve the special case that F(v) = Fo, a constant, and comment on your result. This method of separation of variables is used again in Problems 2.8 and 2.9.
2.7 There are certain simple one-dimensional problems where the equation of motion (Newton's second law) can always be solved, or at least reduced to the problem of doing an integral. One of these (which we have met a couple of times in this chapter) is the motion of a one-dimensional particle subject to a force that depends only on the velocity v, that is, F = F(v). Write down Newton's second law and separate the variables by rewriting it as m dv/F(v) = dt. Now integrate both sides of this equation and show that "S" Vo dv' F (v') t=m Provided you can do the integral, this gives t as a function of v. You can then solve to give v as a function of t. Use this method to solve the special case that F(v) = Fo, a constant, and comment on your result. This method of separation of variables is used again in Problems 2.8 and 2.9.
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Transcribed Image Text:2.7 There are certain simple one-dimensional problems where the equation of motion (Newton's
second law) can always be solved, or at least reduced to the problem of doing an integral. One of these
(which we have met a couple of times in this chapter) is the motion of a one-dimensional particle subject
to a force that depends only on the velocity v, that is, F = F(v). Write down Newton's second law and
separate the variables by rewriting it as m dv/F(v) = dt. Now integrate both sides of this equation and
show that
t = m
v dv'
F(v¹)
Vo
Provided you can do the integral, this gives t as a function of v. You can then solve to give v as a
function of t. Use this method to solve the special case that F(v) = Fo, a constant, and comment on
your result. This method of separation of variables is used again in Problems 2.8 and 2.9.
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