Evaluating time taken

I was trying to create a problem yesterday, something like this:

The yellow region represents a smooth surface, while brown represents the rough one.

A block of mass $m$ enters the rough region while initially traversing with a constant velocity $v$ . We need to find the time taken by the block to come to rest.

The coefficient of friction between the block and the rough surface varies as $k=bx$ , where $x$ is the distance traversed by the block on the rough surface (assuming the intersection of the axes looking lines as the origin) and $b$ is a positive constant.

As an initial approach, I tried to find out the stopping distance of the block, which goes like this.

$\displaystyle mv\frac { dv }{ dx } =-kmg$ or $mv\frac { dv }{ dx } =-bxmg$ as the only force acting on the block is friction.

$\displaystyle mvdv=-bmgxdx\\ m\int _{ v }^{ 0 }{ vdv= } -bmg\int _{ 0 }^{ { x }_{ 0 } }{ xdx }$

Giving

$\displaystyle \frac { { v }^{ 2 } }{ 2 } =\frac { bg{ { x }_{ 0 } }^{ 2 } }{ 2 } \\ \\ { x }_{ 0 }=\frac { v }{ \sqrt { bg } }$

But couldn't proceed on to evaluate time of motion. Please help me, thanks.

PS: Please rectify if there's any mistake in evaluating stopping distance.

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
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Comments

No its absolutely correct.Infact the time would be $t=π/(2√bg)$ .Good to see you make your own prob.(☺☺)

Spandan Senapati - 4 years, 3 months ago

Thanks :)

Swapnil Das - 4 years, 3 months ago

A Former Brilliant Member - 4 years, 3 months ago

Jai Ho :P

The mathematics was complicating for me, I believe, Thanks for the same. Had you encountered such type of question earlier?

not same but similar which i solved by other method ... first time using it :)

@Spandan Senapati

@shubham dhull

Can you tell where you are exactly stuck......BTW here we need to make use of $integral dx/√(a^2-x^2)=arcsin(x/a)+c$

clr or you want me to show it completely how to do ?

Show bhai please :P

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$