# The hardest binomial expansion question i could find help

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#1
Keep in mind this is the hardest question I could find for A-Level maths. But I got so impossibly stuck. I'd be shocked if this isn't A* level but I have no idea for what to do from b) onwards. I know student room is used for hints but I was wondering if someone could work out the answer and then work through how they got it. Thanks!!

18. In special relativity the energy of an object with mass m and speed v is given by:

E = (mc^2)/(1-(v^2)/c^2)^0.5
where c = 3 x 10^8 ms^-1 is the speed of light.

a) Find the first three non-zero terms of the binomial expansion, in increasing powers of v, stating the range of which it it valid.
b) Let E2 be the expansion containing two terms and E be the expansion containing three terms. By what percentage if E3 bigger than E2 if v is:
i) 10% of the speed of light
ii) 90% of the speed of light.
c) Prove E > E3 > E2
0
2 years ago
#2
(Original post by psycholeo)
Keep in mind this is the hardest question I could find for A-Level maths. But I got so impossibly stuck. I'd be shocked if this isn't A* level but I have no idea for what to do from b) onwards. I know student room is used for hints but I was wondering if someone could work out the answer and then work through how they got it. Thanks!!

18. In special relativity the energy of an object with mass m and speed v is given by:

E = (mc^2)/(1-(v^2)/c^2)^0.5
where c = 3 x 10^8 ms^-1 is the speed of light.

a) Find the first three non-zero terms of the binomial expansion, in increasing powers of v, stating the range of which it it valid.
b) Let E2 be the expansion containing two terms and E be the expansion containing three terms. By what percentage if E3 bigger than E2 if v is:
i) 10% of the speed of light
ii) 90% of the speed of light.
c) Prove E > E3 > E2
OK so I'm sure you got the following for part (a) So, then we have  Now if , then  So determine the % increase.
0
#3
Not sure if that's correct, think E = mc^2 + 0.5mv^2 + (3mv^4)/8c^2. And I've tried to work out the percentage increase, E3/E2. I ended up with 1 + (6v^4)/(8c^2v^2 + 8c^4), but wasn't sure how to simplify further
0
2 years ago
#4
(Original post by psycholeo)
Not sure if that's correct, think E = mc^2 + 0.5mv^2 + (3mv^4)/8c^2.
Yeah you're right, I just copy and pasted your formula into Wolfram Alpha and it decided to slap on an extra at the end for some reason.
Post amended, barely anything changes.

And I've tried to work out the percentage increase, E3/E2. I ended up with 1 + (6v^4)/(8c^2v^2 + 8c^4), but wasn't sure how to simplify further
Well I'm not sure how you've arrived at that nonsense. The v's shouldn't even be there -- look at what I did.

EDIT: OK, I suspect you did the division before the substitution. In which case, 0
#5
How did you get the last part? I got an 8 instead of a 4 on the bottom
0
2 years ago
#6
(Original post by psycholeo)
How did you get the last part? I got an 8 instead of a 4 on the bottom
You didn't multiply correctly through.

Take the big fraction after the 2nd equality sign, and multiply the numerator and denominator through by 0
#7
Oh right my bad I got the b) I) correct but what did you get for b) ii)?
0
2 years ago
#8
(Original post by psycholeo)
Oh right my bad I got the b) I) correct but what did you get for b) ii)?
Approx. 17.5%
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#9
(Original post by RDKGames)
Approx. 17.5%
That's what I got but the textbook says 17.7
0
2 years ago
#10
(Original post by psycholeo)
That's what I got but the textbook says 17.7
Seem close enough to me.
0
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