Linear Motion

[pallid matrix]

Harry Hopeless averages 40 km/h for the first half of his car trip. How fast should he drive on the remaining half of the trip to have an average speed of 80 km/h for his trip?

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[waxen oracle]

I got undefined

[pallid matrix]

Really?

[waxen oracle]

yes

[waxen oracle]

d = total distance
v = speed for 2nd half
time for 1st half = (d/2)/40 = d/80
time for 2nd half = (d/2)/v = d/2v

total distance/total time = average speed
d/((d/80)+(d/2v)) = 80
d+(40d/v) = d
40d/v = 0
d/v = 0
d = v×0
v = d/0 = undefined

[rapid mason]

if half of the car trip means half the time, then this has a reasonable answer
the velocities simply average with the weights being the proportions of time

and in this case the weights are equal

so 1/2 v1 + 1/2 v2 = average v

[upbeat quest]

average speed$(\overline{v})=\frac{{v_1}+{v_2}}{2}$

$\therefore \frac{40+v_2}{2}=80$

$\therefore 40+v_2=160$

$\therefore v_2=120$

[gritty birch]

𝔛𝔦𝔩𝔞𝔪

[pallid matrix]

This is a different case. It not moving with a constant acceleration.

[pallid matrix]

What does the 2v mean?

[waxen oracle]

no average speed can't be calculated that way

[pallid matrix]

It can but this case should not be considered.

[waxen oracle]

v is the speed for the 2nd half

2v is from (d/2)/v = d/2v

[pallid matrix]

But if you look careful, it should rather be (d/2)/v=dv/2

[waxen oracle]

what

no

[pallid matrix]

Yeah.

[waxen oracle]

multiply by 2/2

[pallid matrix]

Just look closely.

[waxen oracle]

that's (d/2)/(1/v)

[pallid matrix]

Yeah

[waxen oracle]

this one's (d/2)/v

it's different

[pallid matrix]

(d/2)/v=(d/2)/(1/v)

They are equal.

[upbeat quest]

how so?

[pallid matrix]

Yeah I told him

[waxen oracle]

of course not, look
both numerator are equal
but the denominator aren't
so they aren't equal
it's like 1/2 and 1/3

that's like saying 2=1/2

[pallid matrix]

Then you can ask someone.

[pallid matrix]

No

That's why I said look closely.

[rapid mason]

it can, it's just if you interpret "second half" as second half of the time

[waxen oracle]

hm yea you're right

[pallid matrix]

Actually it is infinity. @here

[pallid matrix]

Oh that's is a different case.

It is not moving with a constant velocity

[frosty ginkgo]

wlog assume the first part to be 20km
he then has used up 30 minutes getting to the half way mark

the whole trip is then 40km and he wants to use a total of 30minutes
so... infinity !?!?!?
i tried 40km and other km values yea for whatever value you put there it is infinity

[pallid matrix]

Which concludes?

[chrome reef]

Yeah, that's correct! It's a trick question. Here's why.
v = (x1 + x2)/(t1 + t2) = (x1 + x2)/(x1/v1 + x2/v2)
Now, suppose x1 = x2 = x/2.
v = 2/(1/v1 + 1/v2)
And this harmonic mean is the key. Let v = 2v1. Then:
2v1 = 2/(1/v1 + 1/v2)
1/v1 = 1/v1 + 1/v2
And now it's easy to see why only v2 = ∞ works. And if we take v > 2v1, then even v2 = ∞ won't be enough.

[chrome reef]

It's a trick question.

[pallid matrix]

OK