Two smooth spheres of masses 4 kg and 2 kg impinge obliquely.
The 2 kg mass is brought to rest by the impact.
(i) Prove that, before impact, they were moving in directions perpendicular to each other.
(ii) Show that, as a result of impact, the kinetic energy gained by the 4 kg mass is...
Homework Statement
The sum of two consecutive squares can be a square: for instance, 32 + 42 = 52
(a) Prove that the sum of m consecutive squares cannot be a square for
the cases m = 3; 4; 5; 6.
(b) Find an example of eleven consecutive squares whose sum is a square.
The Attempt at a...
Ah ok cool, I had tried factoring and leaving a seven outside too before going back and corssing out my work, but I think be both came to the same conclusion. Many thanks for your help dibyendu!
sorry, yes, you're completely right, i must have made a mistake. Ok so is this logic ok?
2S2 ≥ 3 + S1(4-S1)
Well we already know that S1≥3 and S2 ≥ 3
So 2S2 ≥ 6
On the other hand, if we wish to maximise the right hand side of the equation, we want S1 to be its minimum, 3, hence the maximum...
Oh yes i can see, it comes from the fact that (x-1)^2 ≥ 0
So logically it seems to me to try prove
4x + 4y + 4z + xy + yz +xz ≥ 15
Applying AM-GM we get :
(4x + 4y + 4z) ≥ 12
So now we have
xy + yz + zx ≥ 3
which is equivalent to
1/x + 1/y + 1/z ≥ 3
Now I am stuck :P
ok i can see how you rearranged the first inequality but i still don't know where to go from there ? ? ? And may i ask where the inspiration came from to arrange the inequality in that way? Sorry, i don't have much experience solving this kind of problem, so any tips would help immensely! Thanks
5. Suppose that x, y and z are positive real numbers such that xyz = 1.
(a) Prove that
27 \leq(1 + x + y)^{2} + (1 + y + z)^{2} + (1 + z + x)^{2}
with equality if and only if x = y = z = 1.
(b) Prove that
(1 + x + y)^{2} + (1 + y + z)^{2} + (1 + z + x)^{2} \leq 3(x + y + z)^{2}
with...
Thanks for the replies, the bit i don't understand is the use of the centripetal force equation though. In the derivation of the centripetal force f = mv^2/r, it was taken that the mass is moving at a uniform speed and it's acceleration vector was towards the centre of the circle, however if the...
ok i kind of see what you mean, but would it would be right to say that this is not uniform circular motion because the angular velocity is not constant?
Homework Statement
A particle of mass m kg lies on the top of a smooth sphere of radius 2 m.
The sphere is fixed on a horizontal table at P.
The particle is slightly displaced and slides down the sphere.
The particle leaves the sphere at B and strikes the table at Q.
Find
(i) the speed...
Hi i found a question in number theory, involving two equations, it goes as follows:
Let p1, p2, p3 and p4 be 4 different prime numbers satisfying the equations
2p1 + 3p2 + 5p3 + 7p4 = 162
11p1 + 7p2 + 5p3 + 4p4 = 162
Find all possible values of p1p2p3p4.
Not knowing what to do, i used the...