Question

2. (30 points) Suppose an object is dropped from rest at height h and falls by gravity to the ground. During the fall, the object experiences a frictional force proportional to its velocity, $f = -kmv$. a.) Write down the differential equation that describes the velocity $v(t)$ of the object as it falls. Determine the limits of $v(t)$ as $t \to 0$ and $t \to \infty$. b.) Solve the differential equation by expanding $v(t)$ in a perturbative series with at least 5 terms. Show that the series gives the expected result for $t \to 0$. c.) Expand the exact solution (as in Eq. 2.29 on p. 62) in Taylor series expansion and show that it corresponds to the series found in part b.). d.) Calculate the change in kinetic and potential energy as the object falls from height h to the ground. Is the total energy conserved? Explain.

          2.
(30 points) Suppose an object is dropped from rest at height h and falls by gravity
to the ground. During the fall, the object experiences a frictional force proportional
to its velocity, $f = -kmv$.
a.) Write down the differential equation that describes the velocity $v(t)$ of the object
as it falls. Determine the limits of $v(t)$ as $t \to 0$ and $t \to \infty$.
b.) Solve the differential equation by expanding $v(t)$ in a perturbative series with at	least 5 terms. Show that the series gives the expected result for $t \to 0$.
c.) Expand the exact solution (as in Eq. 2.29 on p. 62) in Taylor series expansion
and show that it corresponds to the series found in part b.).
d.) Calculate the change in kinetic and potential energy as the object falls from
height h to the ground. Is the total energy conserved? Explain.

2.
(30 points) Suppose an object is dropped from rest at height h and falls by gravity
to the ground. During the fall, the object experiences a frictional force proportional
to its velocity, f = -kmv.
a.) Write down the differential equation that describes the velocity v(t) of the object
as it falls. Determine the limits of v(t) as t → 0 and t →∞.
b.) Solve the differential equation by expanding v(t) in a perturbative series with at least 5 terms. Show that the series gives the expected result for t → 0.
c.) Expand the exact solution (as in Eq. 2.29 on p. 62) in Taylor series expansion
and show that it corresponds to the series found in part b.).
d.) Calculate the change in kinetic and potential energy as the object falls from
height h to the ground. Is the total energy conserved? Explain.

Added by Douglas W.

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