Starting only with a=(v−u)/t and the definition of average velocity, derive s=ut+½at² and v²=u²+2as. Justify using (u+v)/2 and state two conditions under which your derivation applies.

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Constant acceleration makes v–t linear, so mean velocity=(u+v)/2. Substitute v=u+at for s=ut+½at²; multiply 2s=(u+v)t by a and use at=v−u to obtain v²=u²+2as. Motion is straight-line and acceleration constant. State constant acceleration and straight-line motion with signed quantities. Explain the straight-line velocity graph or equal velocity increments, giving mean (u+v)/2. Use s=mean velocity × t. Show the substitution and expansion giving ut+½at². Show 2as=(u+v)(v−u). Expand the difference of squares and rearrange.

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Starting only with a=(v−u)/t and the definition of average velocity, derive s=ut+½at² and v²=u²+2as. Justify using (u+v)/2 and state two conditions under which your derivation applies.

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