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.

Note by Swapnil Das
1 year, 7 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} \( \boxed{123} \)

Comments

Sort by:

Top Newest

Brilliant Member - 1 year, 7 months ago

Log in to reply

Jai Ho :P

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

Swapnil Das - 1 year, 7 months ago

Log in to reply

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

Brilliant Member - 1 year, 7 months ago

Log in to reply

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

Spandan Senapati - 1 year, 7 months ago

Log in to reply

Thanks :)

Swapnil Das - 1 year, 7 months ago

Log in to reply

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

Brilliant Member - 1 year, 7 months ago

Log in to reply

Show bhai please :P

Swapnil Das - 1 year, 7 months ago

Log in to reply

@Spandan Senapati

@shubham dhull

Swapnil Das - 1 year, 7 months ago

Log in to reply

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

Spandan Senapati - 1 year, 7 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...