3. Classical Mechanics. The differential equation for the velocity v of an object of mass m, restricted to vertical motion and subject only to the forces of gravity and air resistance, is dv m -mg - yv. dt n Eq. (i) we assume that the drag force, -yu where y > 0 is a drag coefficient, is proportional to he velocity. Acceleration due to gravity is denoted by g. Assume that the upward direction is Dositive. (a) Show that the solution of Eq. (i) subject to the initial condition v(0) = vo is mg mg V= vo+ e-rt/m Y (b) Sketch some integral curves, including the equilibrium solution, for Eq. (i). Explain the physical significance of the equilibrium solution. (c) If a ball is initially thrown in the upward direction so that vo > 0, show that it reaches its naximum height when m Yvo t = tmax -In 1+ Y mg =
3. Classical Mechanics. The differential equation for the velocity v of an object of mass m, restricted to vertical motion and subject only to the forces of gravity and air resistance, is dv m -mg - yv. dt n Eq. (i) we assume that the drag force, -yu where y > 0 is a drag coefficient, is proportional to he velocity. Acceleration due to gravity is denoted by g. Assume that the upward direction is Dositive. (a) Show that the solution of Eq. (i) subject to the initial condition v(0) = vo is mg mg V= vo+ e-rt/m Y (b) Sketch some integral curves, including the equilibrium solution, for Eq. (i). Explain the physical significance of the equilibrium solution. (c) If a ball is initially thrown in the upward direction so that vo > 0, show that it reaches its naximum height when m Yvo t = tmax -In 1+ Y mg =
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Transcribed Image Text:8. Classical Mechanics. The differential equation for the velocity v of an object of mass m,
restricted to vertical motion and subject only to the forces of gravity and air resistance, is
dv
(1)
m
-mg - yv.
dt
In Eq. (i) we assume that the drag force, -yu where y> 0 is a drag coefficient, is proportional to
the velocity. Acceleration due to gravity is denoted by g. Assume that the upward direction is
positive.
(a) Show that the solution of Eq. (i) subject to the initial condition v(0) = vo is
mg
mg
V= vo +
mo) e
e-rt/m
Y
(b) Sketch some integral curves, including the equilibrium solution, for Eq. (i). Explain the
physical significance of the equilibrium solution.
(c) If a ball is initially thrown in the upward direction so that vo > 0, show that it reaches its
maximum height when
m
Yvo
t = tmax
(1 +
Y
mg
-In
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