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M2 Projectiles

I'm given the equation x2tan⁡2θ−2axtan⁡θ+(x2−2ab)=0x^2\tan^2 \theta - 2ax\tan \theta + (x^2 - 2ab) = 0.
From this I'm meant to show that the maximum value of xx as θ \theta varies is a(a+2b)\sqrt {a(a+2b)} and that this is achieved when tan⁡θ=aa+2b \tan \theta = \sqrt {\frac {a}{a+2b}} .

I get that the maximum value of x x occurse when θ=45 \theta = 45 , giving me the quadratic equation x2−ax−ab=0x^2 - ax - ab = 0 . Solving this quadratic gives me 2x=a±a(a+4b) 2x = a \pm \sqrt {a(a+4b)} , which kinda looks like what I'm aiming for, but it isn't quite right. Can anyone tell me what I should be doing?

Thanks for any help.

ETA: This isn't the full question - the first part is where the projectiles element is, and all it asks is to derive the first equation given above (which wasn't a problem).
(edited 11 years ago)
Reply 1
Avatar for Bric
Bric
For the first bit, there may be better ways, but you can differentiate the expression w.r.t theta. Then as dx/dtheta = 0 at max, 3 of the terms go away, leaving a nice tan theta = a/x.

Now sub this tan theta back into the original expression to get what you need.

Then for the next bit, sub this value of x back into the tan theta = a/x bit (found earlier) and you're there.

Bric
Reply 2
Avatar for TenOfThem
TenOfThem
18
Original post by Bric
For the first bit, there may be better ways, but you can differentiate the expression w.r.t theta. Then as dx/dtheta = 0 at max, 3 of the terms go away, leaving a nice tan theta = a/x.

Now sub this tan theta back into the original expression to get what you need.

Then for the next bit, sub this value of x back into the tan theta = a/x bit (found earlier) and you're there.

Bric


You can get the tan = a/x by completing the square
Reply 3
Avatar for Julii92
Julii92
OP
Original post by Bric
For the first bit, there may be better ways, but you can differentiate the expression w.r.t theta. Then as dx/dtheta = 0 at max, 3 of the terms go away, leaving a nice tan theta = a/x.

Now sub this tan theta back into the original expression to get what you need.

Then for the next bit, sub this value of x back into the tan theta = a/x bit (found earlier) and you're there.

Bric


Thanks, seems I'll have to complete C3 and C4 before tackling questions like this.
Reply 4
Avatar for Jammy4410
Jammy4410
11
Original post by Julii92
Thanks, seems I'll have to complete C3 and C4 before tackling questions like this.


On the spec, it says prior knowledge is C1/C2/C3/M1
I tried M2 but gave up after I saw trig identities.
Reply 5
Avatar for Julii92
Julii92
OP
Original post by Jammy4410
On the spec, it says prior knowledge is C1/C2/C3/M1
I tried M2 but gave up after I saw trig identities.


I think it says C4 aswell - for OCR in any case.

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