Mechanics vertical resisted motion

[spring glacier]

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[rugged blade]

The goal is to get u > w, where w>0 is the speed of the particle when it lands.

[dusty trench]

follows from solving the ODE no?

the ODE gives you v(t), you're showing this is a decreasing function

since you get $0<T\to v(0)>v(T)$

[hollow burrow]

Omegabet_

[rugged blade]

Well thats trivial for velocity, but the question is asking |v(0)| > |v(T)|

[dusty trench]

so compute them

[rugged blade]

hmm how

[dusty trench]

You can find v(t)

so you know v(0) and v(T)

[rugged blade]

ive tried that stuff here but it didnt work out... i dont think there's anything wrong with my algebra (hopefully)

i can't get an expression for v(T) - which i denote as w in the image

[dusty trench]

energy methods are going to be the easiest probably

[rugged blade]

😔 im sure its doable without it, cos this problem was found in a piece of homework part of a course which does not have energy method in the syllabus

theres also this fact that the total time is a gravity weighted average of the two speeds

$T=\dfrac{u+w}{g}$

[hollow burrow]

@rugged blade

[dusty trench]

Id have to see what you have access to to make decisions

[rugged blade]

but i havent had success

[dusty trench]

also introduce notation properly as a general rule of thumb

at no point did you define what w was

[rugged blade]

sorry, w is just the speed of impact

refer to original image for better detail

[dusty trench]

Im referring to the original

you said v=-w, that's the closest to a definition of w

you also assumed drag is always perfectly parallel to gravity when it changes at the apex

[rugged blade]

just looking if there's a solution thats just using this elementary algebra and calculus

[dusty trench]

k

again

I have no clue what you have access to to do this homework

[rugged blade]

its part of a high school course which has no access to physics formulas

just calculus and algebra

[dusty trench]

that means nothing

'part of a high school course' is nowhere near a workable definition of what content is/isnt available

[rugged blade]

method of this solution not accepted as the course doesnt not assume those formulae
https://math.stackexchange.com/questions/4734237/prove-that-if-air-resistance-is-proportional-to-velocity-then-impact-speed-will

any single variable calculus techniques is accepted

[dusty trench]

[rugged blade]

just use common sense? lmao

[dusty trench]

no fucking shit that's not accepted

youve managed to say no energy method

youve still yet to say what is/isnt allowed

[rugged blade]

bruh

[dusty trench]

dont bruh me cause you cant read

[rugged blade]

literally

anything elementary

is accepted

[dusty trench]

define elementary.

cause energy is pretty elementary, yet not allowed.

[rugged blade]

It's a high school maths course with no assumptions of physics knowledge

[dusty trench]

gl

[rugged blade]

the students only know single variable calc

lmao

how do there exist people with no common sense 😂