I don't know if this is the forum to write this, but I would just like to give firstly some encouragement to the moderators of Physics Forums : you are doing a great job! This is a remarkable resource.
I have subscribed to several of the RSS feeds for the forums, which I find very handy...
oof...
3^2n = 1 mod 8 means that the remainder of 3^2n when divided by 8 is 1. It is another way of saying that 3^2n - 1 is a multiple of 8 (3^2n - 1 = 0 mod 8).
3^2(n+1) = 3^(2n+2) = (3^2n)*(3^2) = (3^2n)*(9) = (3^2n)*1 mod 8
But since you haven't done modulus arithmetic, you probably...
To prove something by induction, for some statement P(n) where n is some positive integer, you must simply do the following:
1/ Show that it is true for k=1, that is P(k) is true.
2/ Show that for each k>=1 that P(k+1) is true.
For the first part, you have already done this:
3^2*1 -1...
Inertia states that a body has the curious property that it naturally stays at the velocity at which it is - velocity is somehow a property of the body, such that it requires something from outside to change it. Thus it naturally stays at rest (velocity = 0) or is moving in a certain direction...
Well, let's start with what you arrived at:
25=25t+-4.91t^2
Now just factor this equation to find the values of t. You probably know the method of completing the squares, that is :
-4.9t^2 + 25t = 25
Dividing through whatever is multiplied on the square term
-t^2 + (25/4.9)t =...
Oh, and I thought I would mention that your problem doesn't actually concern Energy Conservation, but rather the conservation of momentum. Kinetic Energy is usually not conserved in non-elastic collisions.
JJ +
Oops - I just noticed that this forum is for precalculus mathematics - sorry for the answer a little too far ahead.
However, as you noted before you see that the velocity gets smaller and smaller - just find out when it equals zero (it will equal zero when it reaches its maximum height) and...
mv = (m + .583m)v'
v/v' = 1.583
or
v'/v = .631(7)
The question asks for Give your answer in units of the initial speed (i.e. as a fraction of v), so you want v'/v not v/v'. In significant units it is .632.
Firstly, some basic equations of motion, which one can derive using integral calculus:
a = -g (the acceleration is in the opposite direction of the initial velocity, that is, downwards)
v = v(0) + a*t (the velocity at any given moment is equal to the initial velocity plus the acceleration...
Bonus Question
[SIZE="4"]Bonus Question !
Here is a bonus question that I thought of while looking at your question. It actually has an interesting answer:
http://ibphysics.org/images/cylinder-problem.jpg
The paper has certain dimensions, height = 21, width = 30.
A cylinder has a certain h, with circles of radius r at the base. Thus, if the radius of the circles are r, the width of the paper (30) must equal:
4*r + h = 30
(4 radii from two circles plus the height of the cylinder = width...
Actually that should read:
cos 2x = 1 - 2 sin^2 x
This second identity allows you to say also:
1 - 2*sin^2 x = 1 - 2*(1 - cos^2x) and thus:
cos 2x = 2*cos^2 x - 1
I guess you really mean to ask: Does the equation :
cos 2x + cos x = 0
Have any solutions? (assuming for...