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Question

Derive following equations for a uniformly accelerated motion :

(i) v=u+at

(ii) S=ut+12at2

(iii)  v2=u2+2aS

where the symbols have their usual meanings.
 


Solution

(1) First equation of Motion:

V = u + at
soln.
Consider a body of mass “m” having initial velocity “u”.Let after time “t” its final velocity becomes “v” due to uniform acceleration “a”.

Now we know that:
Acceleration = change in velocity/Time taken

=> Acceleration = Final velocity-Initial velocity / time taken

=> a = (v-u) /t
=>at = v-u

=>v = u + at
This is the first equation of motion.
—————————————-
(2) Second equation of motion:
straight s equals ut plus 1 half at squared

sol.
Let the distance travelled by the body be “s”.
We know that
Distance = Average velocity X Time

Also, Average velocity = (u+v)/2

.: Distance (t) = fraction numerator u plus v over denominator 2 end fraction cross times t…….eq.(1)

Again we know that:
v = u + at
substituting this value of “v” in eq.(1), we get

straight s equals fraction numerator straight u plus straight u plus at over denominator 2 end fraction cross times straight t straight s equals fraction numerator 2 straight u plus at over denominator 2 end fraction cross times straight t straight s equals fraction numerator 2 ut plus at squared over denominator 2 end fraction straight s equals fraction numerator 2 ut over denominator 2 end fraction plus at squared over 2

straight s equals ut plus 1 half at squared

This is the 2nd equation of motion.
……………………………………………………………
(3) Third equation of Motion
straight v squared equals straight u squared plus 2 as

sol.
We know that
V = u + at
=> v-u = at
or t = (v-u)/a ………..eq.(2)

Also we know that
Distance = average velocity X Time
.straight s equals fraction numerator straight v plus straight u over denominator 2 end fraction cross times fraction numerator straight v minus straight u over denominator straight a end fraction

straight s equals fraction numerator straight v squared minus straight u squared over denominator 2 straight a end fraction 2 as equals straight v squared minus straight u squared

straight v squared equals straight u squared plus 2 as

This is the third equation of motion

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