**A car accelerates from rest at a constant rate α (alpha) for some time. After which it decelerate at a constant rate β (beta) to come to rest . If total time elapsed is t second then calculate total distance travelled by the car in term of α.β**

Let ‘t_{1}’ be the time for which it accelerates, ‘v’ be the velocity after this time. Now, when the acceleration is α.

v = 0 + αt_{1}

=> v = αt_{1}

=> t_{1} = v/α

Then it decelerates at rate β and stops at t_{2}.

So,

0 = v – βt_{2}

=> v = βt_{2}

=> t_{2} = v/β

Now, t = t_{1} + t_{2} = [v/α] + [v/β] = v[(α+β)/αβ]

=> v = t[αβ/(α+β)]

Now,

Distance travelled in time t_{1} is found as,

v^{2} = 0^{2} + 2αS_{1}

=> S_{1} = v^{2}/(2α)

Distance travelled in time t_{1} is found as,

0^{2} = v^{2} - 2βS_{2}

=> S_{2} = v^{2}/(2β)

Total distance travelled is,

S = S_{1} + S_{2}

=> S = v^{2}/(2α) + v^{2}/(2β)

=> S = [v^{2}/2][1/α + 1/β]

=> S = [v^{2}/2] [(α+β)/αβ]

=> S = ½ [t[αβ/(α+β)]]^{2}[(α+β)/αβ]

=> S = ½ t[αβ/(α+β)]

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