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(A) Derive the formula: \[s=ut+\dfrac{1}{2}a{{t}^{2}}\], where the symbols have usual meaning.

Last updated date: 24th Jul 2024
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Hint: In ordered to answer the first part of the question we would assume some state of a body to prove the above formula. The 1st equation of motion helps us to derive the above formula.

Complete step-by-step solution:
(A) Let us assume a body has an initial velocity = \[u\],
Uniform acceleration = \[a\],
Time= \[t\],
Final velocity= \[v\],
Distance travelled by the body= \[s\],
Now calculate the average velocity \[\Rightarrow \dfrac{InitialVelocity+FinalVelocity}{2}\]
                                                                \[\Rightarrow \dfrac{u+v}{2}\]
And Distance travelled=Average velocity × time
Therefore, \[s\Rightarrow \left( \dfrac{u+v}{2} \right)\times t\]
We know the first equation of motion, \[v=u+at\]
Now we are putting all the above value in equation ,–
  & s\Rightarrow \left( \dfrac{u+u+at}{2} \right)\times t \\
 & s\Rightarrow \left( \dfrac{2ut+a{{t}^{2}}}{2} \right) \\
 & \\
 & \\
\[s=ut+\dfrac{1}{2}a{{t}^{2}}\] where,
initial velocity = \[u\],
Uniform acceleration = \[a\],
Time= \[t\],
Final velocity= \[v\],

Note: Uniform acceleration remains constant with respect to the time . There are some examples of uniform accelerated motion –i.e. dropping a ball from the top, car going along a straight road, skydiver jumping out of the plane.