Derived equation of motion by two method
Answers
Answer:
Derivation of First Equation of Motion
The first equation of motion is:
v = u + at
Derivation of First Equation of Motion by Algebraic Method
It is known that the acceleration (a) of the body is defined as the rate of change of velocity.
So, the acceleration can be written as:
a = v − ut
From this, rearranging the terms, the first equation of motion is obtained, which is:
v = u + at
Derivation of First Equation of Motion by Graphical Method
Consider the diagram of the velocity-time graph of a body.
Derivation Of Equation Of Motion
v = at + u
Derivation of First Equation of Motion by Calculus Method
It is known that,
Derivation Of Equation Of Motion
So,
Derivation Of Equation Of Motion
Derivation of Second Equation of Motion
The second equation of motion is:
S = ut + ½ a2
Derivation of Second Equation of Motion by Algebraic Method
Consider the same notations for the derivation of the second equation of motion by simple algebraic method.
Derivation Of Second Equation Of Motion
Derivation of Second Equation of Motion by Graphical Method
Taking the same diagram used in first law derivation:
Derivation Of Equation Of Motion
In this diagram, the distance travelled (S) = Area of figure OABC = Area of rectangle OADC + Area of triangle ABD.
Now, the area of the rectangle OADC = OA × OC = ut
And, Area of triangle ABD = (1/2) × Area of rectangle AEBD = (1/2) at2 (Since, AD = t and BD = at)
Thus, the total distance covered will be:
S = ut + (1/2) at2
Derivation of Second Equation of Motion by Calculus Method
Velocity is the rate of change of displacement.
Mathematically, this is expressed as
v=dsdt
Rearranging the equation, we get
ds=vdt
Substituting the first equation of motion in the above equation, we get
ds=(u+at)dt =(udt+atdt) ∫s0ds=∫t0udt+∫t0atdt s=ut+12at2
Derivation of Third Equation of Motion
The third equation of motion is:
v2 = u2 + 2aS
Derivation of Third Equation of Motion by Algebraic Method
Derivation Of Third Equation Of Motion
Derivation of Third Equation of Motion by Graphical Method
Derivation Of Equation Of Motion
The total distance travelled, S = Area of trapezium OABC.
So, S= 1/2(SumofParallelSides)×Height
S=(OA+CB)×OC
Since, OA = u, CB = v, and OC = t
The above equation becomes
S= 1/2(u+v)×t
Now, since t = (v – u)/ a
The above equation can be written as:
S= 1/2(u+v)×(v-u)/a
Rearranging the equation, we get
S= 1/2(v+u)×(v-u)/a
S = (v2-u2)/2a
Third equation of motion is obtained by solving the above equation:
v2 = u2+2aS
Derivation of Third Equation of Motion by Calculus Method
It is known that,
Derivation Of Equation Of Motion
These were the detailed derivations for equations of motion in the graphical method, algebraic method and calculus method.
Explanation:
Answer:
First Equation of Motion : v = u + a t.
Second Equation of Motion : s = u t + 1 2 a t 2.
Third Equation of Motion : v 2 = u 2 + 2 a s.
Explanation: