# Equations of Motion

Helena Dedic

## Derivation of Equations of Motion

Consider a v-t graph for an object moving with constant velocity.

• Motion with constant velocity implies that the acceleration is equal to zero and therefore the v - t graph is a horizontal line.

${\displaystyle v(t)=v_{o}}$


• The displacement is the area under the v - t graph

We write ${\displaystyle \Delta x(t)=v_{o}t}$

• Motion with constant acceleration implies that the velocity is a linear function of time. Given that the initial velocity is ${\displaystyle v_{0}}$ we can write

${\displaystyle v(t)=v_{o}+at}$


where the acceleration is the slope of the graph and${\displaystyle v_{o}}$ is the intercept.

• The displacement is the area under the v - t graph

We write

${\displaystyle \Delta x(t)=v_{o}t+{\frac {1}{2}}at^{2}}$


• The displacement in both cases is ${\displaystyle \Delta x(t)=x(t)-x_{o}}$

• There are two independent equations describing motion with constant acceleration and five variables: ${\displaystyle \Delta x(t)}$, v(t), ${\displaystyle v_{0}}$, a and t. Consequently, in any problem we can solve for two of those variables and three other must be given.

• We can derive another useful equation by eliminating t from the two equations above.

We begin with equation

${\displaystyle v(t)=v_{o}+at}$


We isolate t:

${\displaystyle t={\frac {v(t)-v_{o}}{a}}}$


Then we substitute for t in the equation for the displacement:

${\displaystyle \Delta x(t)=v_{o}\left({\frac {v(t)-v_{o}}{a}}\right)+{\frac {1}{2}}a\left({\frac {v(t)-v_{o}}{a}}\right)^{2}}$


We will now multiply both sides by 2a. This step will eliminate fractions from this equation.

${\displaystyle 2a\Delta x(t)=2av_{o}\left({\frac {v(t)-v_{o}}{a}}\right)+2a{\frac {1}{2}}a\left({\frac {v(t)-v_{o}}{a}}\right)^{2}}$


${\displaystyle 2a\Delta x(t)=2v_{o}(v(t)-v_{o})+(v(t)-v_{o})^{2}}$


Expanding:

${\displaystyle 2a\Delta x(t)=2v_{o}v(t)-2v_{o}^{2}+v^{2}(t)+v_{o}^{2}-2v_{o}v(t)}$


${\displaystyle 2a\Delta x(t)=-v_{o}^{2}+v^{2}(t)}$


This can finally be written in the form:

${\displaystyle v^{2}(t)=v_{o}^{2}+2a\Delta x(t)}$