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Uniform and Non Uniform

A car acceler...

Question

A car accelerates from rest at a constant rate a for some time after which it decelerates at constant rate to come to rest. If the total time elapsed is ts. What will be the total distance travelled?

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Solution

Dear Student,
Given,
A car accelerates from rest (u = 0) at a constant rate $a$ for some time say t₁ and reaches a velocity v₁ (say).
Let the distance travelled while accelerating be x₁. Then
$x subscript 1 equals u t subscript 1 plus 1 half a t subscript 1 squared x subscript 1 equals 1 half a t subscript 1 squared space open parentheses because u equals 0 close parentheses A l s o comma v subscript 1 equals u plus a t subscript 1 v subscript 1 equals a t subscript 1$
Then it decelerates from v₁ at a constant rate $a apostrophe$ for some time say t₂to come to rest (ie, v = 0).
Let the distance travelled while decelerating be x₂. Then
$x subscript 2 equals v subscript 1 t subscript 2 minus 1 half a apostrophe t subscript 2 squared A l s o comma v squared equals v subscript 1 squared minus 2 a apostrophe x subscript 2 v subscript 1 squared equals 2 a apostrophe x subscript 2 space left parenthesis because f i n a l space v e l o c i t y comma space v space equals space 0 right parenthesis$
Total time taken is t
Then,
$t equals t subscript 1 plus t subscript 2$
Let the total distance travelled be x
$x equals x subscript 1 plus x subscript 2$
$x equals x subscript 1 plus x subscript 2 equals 1 half a t subscript 1 squared plus v subscript 1 t subscript 2 minus 1 half a to the power of apostrophe t subscript 2 squared p u t t i n g space t h e space v a l u e space o f space v subscript 1 equals a t subscript 1 comma space w e space g e t x equals 1 half a t subscript 1 squared plus a t subscript 1 t subscript 2 minus 1 half a to the power of apostrophe t subscript 2 squared A l s o comma t subscript 2 equals t minus t subscript 1 w e space g e t x equals 1 half a t subscript 1 squared plus a t subscript 1 open parentheses t minus t subscript 1 close parentheses minus 1 half a to the power of apostrophe open parentheses t minus t subscript 1 close parentheses squared T h u s comma v a l u e s space o f space a comma space a apostrophe space a n d space t subscript 1 space i s space r e q u i r e d space t o space f i n d space t h e space t o t a l space d i s tan c e space t r a v e l l e d.$
Regards

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