Deriving equations



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=Δd/Δt; Vav=(vi+vf)/2 Δd= change in distance; Δt = change in time, Vav= average velocity

aav=Δv/Δt Δv= change in velocity; Δt=change in time

Given these definitions, it is possible to derive the fundamental kinematics equations:

vf = vi + aΔt

Δd= viΔt + aΔt2

vf=)

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 (Δt).

vf = vi + aΔt

a=(vf-vI)/ Δt (by definition)

(solve for vf)

(2) Express the change in distance (Δd) as a function of the initial velocity (vi), change in time (Δt), and acceleration (a) (Not in terms of final velocity vf).

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=Δd/Δt) and solve for Δd.

Δd= viΔt + aΔt2

(3) Express the final velocity in terms of the initial velocity (vi), acceleration (a ), and the change in distance (Δd). (Not in terms of time Δt!)

Hint: Solve the first equation for Δt, subsitute this in the second equation, and solve for vf.

vf=

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