Here are the steps involved in solving a conservation of energy
problem.

State what's given and what you
are to find.

Identify your system and specifically list
what objects are included in the system.

List the forces external to the system.
Indicate whether or not these forces do positive, zero, or negative work
on the system.

Conservation of energy problems always deal
with a beginning point and an ending point. These are termed the initial and final
states. Decide what you will use for these states and list those choices. Then, in your solution, label related quantities with the subscripts i and f for initial and final.

Identify the forms of energy in the system
and how the energy terms change from the initial to the final state. Is ΔK positive, negative, or zero? How about
ΔU_{g} and
ΔU_{e}?

Draw a diagram in which you show the
initial and final states and specify the
positive axis directions and the origin. Choose the origin this way:
a) If there's an elastic potential energy change in the problem, pick
the origin at the relaxed position of the spring. This will ensure that
the elastic potential energy of the relaxed spring is 0. b) If the only
potential energy change is gravitational, the choice of origin is
arbitrary but is often selected to be the lowest point. Select +y to be up, as this will help you avoid sign difficulties later.

Write the general
conservation of energy equation, W_{ext} = ΔE_{sys}.

Apply what you know to the solution of
the problem. Substitute terms for
initial and final energy changes on the righthand side of the equation.
Then expand the energy changes in terms of initial and final terms: K_{f},
K_{i}, U_{gf}, U_{gi}, U_{ef}, and U_{ei}.
Solve for the unknown symbolically.

Apply the usual checks of signs, units,
and sense.

Substitute values and units and reduce.

Problem: Consider a block sliding down a frictionless,
inclined plane. The problem is to find the speed v_{f} of the box of mass m after the box has slid a particular
distance, d, along the plane. We'll assume the box is moving
initially with speed, v_{i}. The angle of the
plane above the horizontal is θ. For this
example, we'll number the steps to correspond to the numbers above.
1. Given: d, m, v_{i}, θ, g
Goal: Find v_{f} in terms of the givens.
2. System: block, Earth
3. External force: normal (does no work)
4. initial state: highest position of block
final state: lowest position of block
5.
ΔU_{g} is negative since the block
decreases in elevation;
ΔK is positive since the block speeds up 
The diagram below shows (among other things):
 the direction of +y (up)
 the lowest position of the block
designated as y_{f}= 0

6. 
7,8. Now we're ready for equations.

We first write the general
conservation of energy law. 
W_{ext} is 0, because
the normal force does no work on the block. 
The energy changes in the system are
kinetic and gravitational. 
Expand the delta notation in terms of
initial and final quantities. 
The final gravitational energy is 0,
because we specified the lowest position of the object to be 0. 
Substitute expressions for the
remaining energy terms. 
Solve for the unknown. 
Use the geometry of the plane to
substitute for y_{i} in terms of given symbols. Select
the positive root since we're only interested in the speed. 
9. Checks: The units reduce to m/s. The sign of the
quantity under the radical is positive, so the square root yields a
real number. K_{f}  K_{i} is positive since v_{f} > v_{i}. U_{gf}  U_{gi}_{ }= mgy_{i} is negative.
10. There are no values to substitute. 
Problem: A ball of mass 1.25 kg is placed in front of a springloaded platform
mounted on a table 0.500 m above the floor as shown to the right. The
spring constant of the spring is 84.5 N/m, and the spring is compressed 0.1435
m. The uncompressed position of the right edge of the spring is at the
right edge of the table. At t = 0, the ball is quickly released. Determine the speed of
the ball after the spring is released and the ball leaves the edge of
the table.
1. Given:
m = 1.25 kg k = 84.5 N/m
x_{i} = 0.1435 m x_{f} = 0 m (Note that x_{i} is
negative due to the way the coordinate system is set up.)
v_{i} = 0 m/s
Goal: Find v_{f}.
2. System: spring, ball (The
Earth and gravity needn't be included because the gravitational
potential energy of the ball doesn't change.)
3. External forces: normal and weight (neither does work)
4. initial state: spring fully compressed
final state: spring fully decompressed
5.
ΔU_{e} is negative since the spring
relaxes;
ΔK is positive since the ball speeds up 
The diagram below shows:
 the direction of +x is the the right
 the position of the relaxed spring is x_{f}= 0

6. 
710. Now for equations...

Write the general
conservation of energy law. 
W_{ext} is 0, because
the normal force and gravity do no work on the block. The energy changes in the system are
kinetic and elastic. 
Expand the delta notation in terms of
initial and final quantities. 
The initial kinetic energy and final
elastic potential energy are 0. 
Substitute expressions for the
remaining energy terms. 
Solve for the unknown. 
Substitute values with units. 
Select the positive root since we're only interested in the speed. 
Checks: The units m[N/(m∙kg)]^{½} reduce to m/s. The sign of the quantity under the radical is positive, so
the square root yields a real number.
Here are more example problems (presented at a WX session). 
