How to Derive Equations of Motion - with & without Calculus - Motion Video Lecture - Class 9

hello my name is Alex and I'm here to
help today I'm going to help you derive
the equations of motion that are
commonly used in physics these are all
assuming constant acceleration
acceleration can equal zero but it
cannot change okay there are three
important things that need to be brought
up before we can get to the equations
and they are position velocity and
acceleration position is basically just
a measurement from point A to point B
and it can be calculated like this
here's an example
he drove 75 miles away from home that
just means his position 75 miles away
relative to his home it's just a
measurement in this case if you do the
unit conversion it's 120 point seven
kilometers velocity is the change in
position over time if you want to figure
out how long it takes to go somewhere
and you know the distance you can
calculate the average velocity and
here's the here's an example
he ran 100 meters in nine point five
eight seconds the velocity equals 100
meters divided by nine point five eight
seconds that equals ten point four three
eight meters per second if you want to
calculate how long it takes velocity to
change over time this is known as
acceleration
now here's an example
a car can go from zero miles per hour to
60 miles per hour in 2.2 seconds
the acceleration is equal to 60 miles
per hour divided by 2.2 seconds doing
unit conversion and the proper
calculation equals twelve point one nine
two meters per second squared
once you get the final definitions down
then you can work on deriving the
equations and I'm going to work
backwards starting with acceleration at
first I'm going to do this using no
calculus later I'll get to the calculus
derivation acceleration is defined as
the change in velocity over the change
in time an equation form that looks like
this
AE equal Delta V over delta T this Delta
sign means change V minus V initial t
minus T initial typically T initial
equals zero we usually start from T
equals zero and most problems unless
otherwise stated so given that 80
bring the tea over to the a equals V
minus V initial and this could be
re-written to our first major equation v
equals V initial plus a times T if you
know the initial velocity and you know
acceleration and you know time you can
calculate V next you move on to the
definition of velocity change in
position over change in time the average
velocity can be calculated as the total
distance traveled over the time it took
to go that distance if you'll also
remember how you find the average of two
numbers one number plus another number
divided by two average velocity final
velocity plus initial velocity divided
by two from here you can get the second
major equation distance traveled equals
the average velocity over multiply by
the time it takes third equation can be
derived by combining the first and
second if you take that first equation
this one here and you plug it in to this
V right here you get Delta X average
velocity times time and then you plug in
the first equation to write here
from here
combining terms it can be simplified to
our third major equation if T initial
equals zero Delta X equals V initial
times time plus 1/2 a
third equation and the first equation
first equation can be rewritten to solve
for time T equals V minus V initial over
a if you plug this into this equation
you get this whoops
delta x equals V naught x that big guy
1/2 a times that big guy again squared
it looks a little messy but when you do
the manipulation and you simplify you
get this this is the form that it's
typically written in v squared equals V
initial squared plus 2 a delta X and
this is known as the time-independent
equation because there is no time part
of to it so if you know the initial
velocity and acceleration in position
you can calculate final velocity without
even knowing how long it took these are
the four major equations that are used
in kinematics and I derived them without
calculus it's possible to do it with
calculus and I'm going to show you that
now
acceleration is defined as the
derivative of velocity with respect to
time rewriting this you can get a DT
equals DV integrating both sides you get
a DT from zero to T DV from VI to V
equals 80 time equals V minus the
initial rewriting for V and you get the
first major equation v equals VI plus 80
and it's the same as this one over here
then next moving to velocity you know
that velocity is defined as the
derivative of position with respect to
time change in position over change in
time rewriting this DX equals V DT and
plugging the first equation plugging
this into here we get VI plus 80 the sum
of that times DT then integrating both
sides we get 1 DX from X I to x equals
VI plus 80 DT from 0 to T integrating
and simplifying we get our other major
equation change a position equals VI
times T plus one-half a t-square the
time independent equation can be derived
using calculus as well going from the
definitions of acceleration and velocity
they can both be rewritten to solve for
DT DT equals DV over a equals DX over V
using these second two fractions here
you have a DX equals V DV integrating
both sides from X I to X and V I V
integrate simplify what doesn't fit
there you have X minus X I times a
equals V squared minus the initial
squared over two rewriting this is the
way it's commonly written P squared
equals the initial squared plus two a
Delta X
okay so those are the three equations
what's very important to realize is that
these the three components we have
position velocity and acceleration are
all vectors which means they have an X a
Y and Z component you won't always see a
Z component but technically it's there
usually it's just zero what if things
are moving in three dimensions you have
to worry about the Z so given the three
equations you have to be mindful when
you talk about velocity position
acceleration that you specify which
component you mean notice the subscripts
V final x the initial X a X you have to
do that for the Y direction and the Z
direction you have V initial Y a Y Delta
Y the initial Z a Z Delta Z vectors have
components very important this set of
equations will govern pretty much all of
the motion in physics assuming constant
acceleration
if you have a problem that you're
struggling with send it over to me Alex
please help calm