Distance constraints the acceleration

Question

A particle starts from rest at !!t = 0!! from the point !!x = 0!!, and comes to rest at !!t = 1!! at the point !!x = 1!!. Instantaneous acceleration is given by a.

Then which of the following is/are true?

  • !!a!! cannot remain positive throughout

  • !!a!! cannot exceed !!2!! at any point in its path

  • !!|a|!! must be equal to !!4!! at some point or points in its path

Solution

Now since the initial and final velocity are !!0!!, so it can not not continuously accelerate or decelerate. Hence first option is correct.

Consider the VT graph where it has a uniform acceleration of 4 and deceleration of 4 for half a second each. The area under the curve be 1 right?

Now consider any graph which has an area of 1. Then it should definitely cuts the given graph otherwise the area can not be 1. And if it cuts that means its slope is more than 4.

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