Need help on a mechanics question (2 Viewers)

tickboom · Nov 18, 2020

Does anybody have any ideas for part ii)? I can show it for n=3, but can't work out how to show it in general. I tried induction but it didn't seem to work for me.

black.mamba · Nov 18, 2020

you already found x in part i), and find t as well

use u<v(av)<v where v=x/t

Drdusk · Nov 18, 2020

tickboom said:
Does anybody have any ideas for part ii)? I can show it for n=3, but can't work out how to show it in general. I tried induction but it didn't seem to work for me.

View attachment 29542

Don't see why you would have to use induction.

Using $a = \dfrac{dv}{dt}$ you get

$t = \dfrac{u^{n-1} - v^{n-1}}{k(n-1)(uv)^{n-1}}$

Calculating avg speed you get

$v_{avg} = \dfrac{x}{t} = \dfrac{u^{n-2} - v^{n-2}}{k(n-2)(uv)^{n-2}}\times \dfrac{k(n-1)(uv)^{n-1}}{u^{n-1} - v^{n-1}}$

$= uv \times\dfrac{a_{n-2}}{a_{n-1}}$

Now we know that v < v_avg < u because the speed is constantly decreasing so it must be less than the start speed but greater than the instantaneous speed.

$\therefore v < uv\times \dfrac{a_{n-2}}{a_{n-1}} < u$

Simplifying further

$\therefore v < \dfrac{a_{n-1}}{a_{n-2}} < u$

black.mamba · Nov 18, 2020

damn just as i was about to upload my photo lol

tickboom · Nov 18, 2020

Drdusk said:
Don't see why you would have to use induction.

Using $a = \dfrac{dv}{dt}$ you get

$t = \dfrac{u^{n-1} - v^{n-1}}{k(n-1)(uv)^{n-1}}$

Calculating avg speed you get

$v_{avg} = \dfrac{x}{t} = \dfrac{u^{n-2} - v^{n-2}}{k(n-2)(uv)^{n-2}}\times \dfrac{k(n-1)(uv)^{n-1}}{u^{n-1} - v^{n-1}}$

$= uv \times\dfrac{a_{n-2}}{a_{n-1}}$

Now we know that v < v_avg < u because the speed is constantly decreasing so it must be less than the start speed but greater than the instantaneous speed.

$\therefore v < uv\times \dfrac{a_{n-2}}{a_{n-1}} < u$

Simplifying further

$\therefore v < \dfrac{a_{n-1}}{a_{n-2}} < u$

Brilliant! Thanks so much!

Trebla · Nov 18, 2020

FYI this question was from this year’s BoS trials

tickboom · Nov 18, 2020

Trebla said:
FYI this question was from this year’s BoS trials

Nice one. It's a great question!

Need help on a mechanics question (2 Viewers)

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