An alternate method to solve many dynamic problems is the energy method. This method is particularly helpful if the problem involves finding position or velocity when two different positions along its motion path are known.
Basically, energy methods analyze all energy added or deleted as an object moves between two points. The difference in energy between the two points must be accounted for with an increase or decrease in kinetic energy (i.e. velocity). In equation form, this is stated as,
T1 + ΣU1-2 = T2
T1 = kinetic energy at position
1 = ½mv12
T2 = kinetic energy at position
2 = ½mv22 ΣU1-2 =
total change in energy (i.e. work)
between positions 1 and 2
Energy methods are a convenient method to solve for velocity or position. However, energy methods do not work for acceleration.
Notice, energy terms are scalars which do not have direction. The energy equation is not a vector equation. This means that only one unknown can be determined.
The motion of an object is also stored energy and is called kinetic energy. This can be illustrated by applying a simple force, F, on an object over a distance, s, and finding its final velocity. In other words, the change in work (FΔs) will produce a change in kinetic energy, or simply
ΔUkinetic = F Δs
dUkinetic = Fds
Using the relationships, "F = ma" and, "a ds = v dv", gives,
dUkinetic = m(v dv/ds) ds = mv dv
Integrating both sides gives,
Ukinetic = ½mv2
Thus the kinetic energy, T, for any object in motion is