# The hardest binomial expansion question i could find help

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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

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

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#2

(Original post by

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

**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

So, then we have

Now if , then

So determine the % increase.

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(Original post by

OK so I'm sure you got the following for part (a)

So, then we have

Now if , then

So determine the % increase.

**RDKGames**)OK so I'm sure you got the following for part (a)

So, then we have

Now if , then

So determine the % increase.

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#4

(Original post by

Not sure if that's correct, think E = mc^2 + 0.5mv^2 + (3mv^4)/8c^2.

**psycholeo**)Not sure if that's correct, think E = mc^2 + 0.5mv^2 + (3mv^4)/8c^2.

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

EDIT: OK, I suspect you did the division

*before*the substitution. In which case,

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How did you get the last part? I got an 8 instead of a 4 on the bottom

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#6

(Original post by

How did you get the last part? I got an 8 instead of a 4 on the bottom

**psycholeo**)How did you get the last part? I got an 8 instead of a 4 on the bottom

Take the big fraction after the 2nd equality sign, and multiply the numerator and denominator through by

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Oh right my bad I got the b) I) correct but what did you get for b) ii)?

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#8

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Oh right my bad I got the b) I) correct but what did you get for b) ii)?

**psycholeo**)Oh right my bad I got the b) I) correct but what did you get for b) ii)?

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(Original post by

Approx. 17.5%

**RDKGames**)Approx. 17.5%

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#10

(Original post by

That's what I got but the textbook says 17.7

**psycholeo**)That's what I got but the textbook says 17.7

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