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 ‘t1’ be the time for which it accelerates, ‘v’ be the velocity after this time. Now, when the acceleration is α.

v = 0 + αt1

=> v = αt1

=> t1 = v/α

Then it decelerates at rate β and stops at t2.

So,

0 = v – βt2

=> v = βt2

=> t2 = v/β

Now, t = t1 + t2 = [v/α] + [v/β] = v[(α+β)/αβ]

=> v = t[αβ/(α+β)]

Now,

Distance travelled in time t1 is found as,

v2 = 02 + 2αS1

=> S1 = v2/(2α)

Distance travelled in time t1 is found as,

02 = v2 - 2βS2

=> S2 = v2/(2β)

Total distance travelled is,

S = S1 + S2

=> S = v2/(2α) + v2/(2β)

=> S = [v2/2][1/α + 1/β]

=> S = [v2/2] [(α+β)/αβ]

=> S = ½ [t[αβ/(α+β)]]2[(α+β)/αβ]

=> S = ½ t[αβ/(α+β)]

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