EQUATION DERIVATION
Introduction: The average physics text introduces more than 100 basic
equations, many of which have one or more alternate expressions. When faced
with learning so many equations, most students resort to rote memorization and
generally fail to comprehend the relationships expressed by such equations.
Regardless of the field of study, it is important to show students how equations are
derived so they can appreciate the logic that is at the core of the discipline. When
students understand this logic, they will be better able to develop problem solving
skills.
Example: Galileo Galilei (1564-1642) is credited with the first clear studies of
acceleration and velocity. His definitions for these values was:
vav=d/t; Vav=(vi+vf)/2 d= change in distance; t = change in time, Vav=
average velocity
a=v/t
v= change in velocity; t=change in time
Given these definitions, it is possible to derive the fundamental kinematics
equations:
vf = vi + at
1
d= vit + 2 at2
vf= vi2+2ad )
Show how these equations are derived. It is important to note which variables are
not in the equation and eliminate them by substitution.
(1) Express the final velocity (vf) in terms of the initial velocity (vi) , acceleration
(a), and the change in time (t).
vf = vi + at
a=(vf-vI)/t (by definition)
(solve for vf)
(2) Express the change in distance (d) as a function of the initial velocity (vi),
change in time (t), and acceleration (a) (Not in terms of final velocity vf).
6.5 Equation Derivation
Hint: Substitute the value of vf from equation 1 into the definition for average
velocity (vav = (vi+ vf)/2), then substitute the new value for vav into the the
devinition for velocity (vav=d/t) and solve for d.
1
d= vit + 2 at2
(3) Express the final velocity in terms of the initial velocity (vi), acceleration (a ),
and the change in distance (d). (Not in terms of time t!)
Hint: Solve the first equation for t, subsitute this in the second equation, and
solve for vf.
vf= vi2+2ad
6.5 Equation Derivation