drive third equation of motion without graph
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Answer:
There are three equations of motion that can be used to derive components such as displacement(s), velocity (initial and final), time(t) and acceleration(a). The following are the three equation of motion:
First Equation of Motion : v=u+at
Second Equation of Motion : s=ut+12at2
Third Equation of Motion : v2=u2+2as
To brush up on the basics of motion, refer the article listed below::
Introduction to Motion
Derivation of Equation of Motion
The equations of motion can be derived using the following methods:
Derivation of equations of motion by Simple Algebraic Method
Derivation of equations of Motion by Graphical Method
Derivation equations of Motion by Calculus Method
In the next few sections, the equations of motion are derived by all the three methods in a simple and easy to understand way.
Derivation of First Equation of Motion
For the derivation, let us consider a body moving in a straight line with uniform acceleration. Then, let the initial velocity be u, acceleration is denoted as a, time period is denoted as t, velocity is denoted as v, and the distance travelled is denoted as s.
Derivation of First equation of Motion by Algebraic MethodDerivation of First equation of Motion by Graphical MethodDerivation of First equation of Motion by Calculus Method
Derivation of First Equation of Motion by Algebraic Method
We know that the acceleration of the body is defined as the rate of change of velocity.
Mathematically, acceleration is represented as follows:
a=v−ut
where v is the final velocity and u is the initial velocity.
Rearranging the above equation, we arrive at the first equation of motion as follows:
v=u+at
Derivation of First Equation of Motion by Graphical Method
The first equation of motion can be derived using a velocity-time graph for a moving object with an initial velocity of u, final velocity v, and acceleration a.

In the above graph,
The velocity of the body changes from A to B in time t at a uniform rate.
BC is the final velocity and OC is the total time t.
A perpendicular is drawn from B to OC, a parallel line is drawn from A to D, and another perpendicular is drawn from B to OE (represented by dotted lines).
Following details are obtained from the graph above:
The initial velocity of the body, u = OA
The final velocity of the body, v = BC
From the graph, we know that
BC = BD + DC
Therefore, v = BD + DC
v = BD + OA (since DC = OA)
Finally,
v = BD + u (since OA = u) (Equation 1)
Now, since the slope of a velocity-time graph is equal to acceleration a,
So,
a = slope of line AB
a = BD/AD
Since AD = AC = t, the above equation becomes:
BD = at (Equation 2)
Now, combining Equation 1 & 2, the following is obtained:
v = u + at
Derivation of First Equation of Motion by Calculus Method
Since acceleration is the rate of change of velocity, it can be mathematically written as:
a=dvdt
Rearranging the above equation, we get
adt=dv
Integrating both the sides, we get
∫t0adt=∫vudv
at=v−u
Rearranging, we get
v=u+at
Derivation of Second Equation of Motion
For the derivation of the second equation of motion, consider the same variables that were used for derivation of the first equation of motion.
Derivation of Second equation of Motion by Algebraic MethodDerivation of Second equation of Motion by Graphical MethodDerivation of Second equation of Motion by Calculus Method
Derivation of Second Equation of Motion by Algebraic Method
Velocity is defined as the rate of change of displacement. This is mathematically represented as:
Velocity=DisplacementTime
Rearranging, we get
Displacement=Velcoity×Time
If the velocity is not constant then in the above equation we can use average velocity in the place of velocity and rewrite the equation as follows:
Displacement=(InitialVelo
Question
drive third equation of motion without graph
Answer
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