show the derivation of all three equations of motion
Answers
Answer:
u - initial velocity ; v - final velocity ; S - distance traveled ;
a - acceleration ; t - time taken ;
We can derive three equations of motion using definitions.
acceleration = change in velocity / time taken
acceleration = ( final velocity - initial velocity ) / time taken
a = ( v - u ) / t
a t = v - u
u + a t = v
v = u + a t
distance traveled = average velocity * time taken
distance traveled = ( ( initial velocity + final velocity ) / 2 ) * time taken
S = ( ( u + v ) / 2 ) * t
From 1. v = u + a t
S = ( ( u + ( u + a t ) ) / 2 ) *t
S = ( ( 2 u + a t ) / 2 ) * t
S = ( ( 2 u / 2 ) + ( a t / 2 ) ) * t
S = ( u + a t / 2 ) * t
S = u t + ( a t^2 ) / 2
2. S = u t + ( a t^2 ) / 2
distance traveled = average velocity * time taken
distance traveled = ( ( initial velocity + final velocity ) / 2 ) * time taken
S = ( ( u + v ) / 2 ) * t
From 1. v = u + a t
v - u = a t
( v - u ) / a = t
t = ( v - u ) / a
S = ( ( u + v ) / 2 ) * ( ( v - u ) / a )
S = ( ( v + u ) / 2 ) * ( ( v - u ) / a )
S = ( ( v + u ) * ( v - u ) ) / 2 a
S = ( v^2 - u ^2 ) / 2 a
2 a S = v^2 - u^2
u^2 + 2 a S = v^2
3. v^2 = u^2 + 2 a S
Thus, all the three equations of motion can be derived by definitions.
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