Forest UCM Energy Line1D
The equation of motion for a system restricted to 1-D is readily solved from conservation of energy when the force is conservative.
- cosntant
The ambiguity in the sign of the above relation, due to the square root operation, is easily resolved in one dimension by inspection and more difficult to resolve in 3-D.
The velocity can change direction (signs) during the motion. In such cases it is best to separte the inegral into a part for one direction of the velocity and a second integral for the case of a negative velocity.
Free fall
Consider a rock dropped at t=0 from a tower of height h.
The potential energy stored in the rock at any instant is given by
- Note
- The potential is highest at x=0 and becomes negative as x increases
The initial total energy is
or
spring example (problem 2.8)
Consider the problem of a mass attached to a spring in 1-D.
The potential is given by
let
- and
then
- = amplitude of oscillating motion