[phoenirius]

A 0.5kg ball is thrown vertically upward at a speed of 25m/s. Its velocity, v(t) (in m/s), at time t(in seconds0, is given by v(t)=25-9.8t. Its kinetic evergy, K(v) (in Joules), is given by K(v)=0.5v^2. Express the balls kinetic energy as a function of time, K(t).

OKay I have no idea even where to start.
Thanks.

[dmatos]

They've given you two formulas:

v(t) = 25 - 9.8t
K(v) = 0.5v^2

Note that for the second formula, it is perfectly acceptable (and correct) to write:

K(t) = 0.5[v(t)]^2

Then, all you have to do is perform those operations:

K(t) = 0.5 * (25 - 9.8t)^2

Expand and simplify for your answer.

[phoenirius]

Okay but to find the function would this be right...

f(x) = 0.5(25-9.8t)^2
And how do I solve?

[dmatos]

First, a quick lesson about functions. A function is defined in terms of variables. A variable is something that can change. When writing a function, it is the variable names that go inside the bracket. Observe:
f(x) = x^2 + 2x + 5
g(m) = 4m + 8
You can also have multivariable functions:
H(a,b,c) = 4a+bc

You have been given a function v(t) = 25-9.8t. This function describes how the velocity of the ball changes as the variable "t" (in this case, time) changes. You also have the function K(v) = 0.5mv^2 (I think you left the m out). This describes how the kinetic energy of the ball changes as the velocity of the ball changes. So far so good?

Well, the question is now asking you how the kinetic energy changes as time changes. What do you have to do? Combine the two functions. On the left hand side, you are going to have something indicating that the function describes how the kinetic energy changes as time changes. Let's call that K(t). What goes on the right hand side?

Well, start with 0.5mv^2. But, this has to be in terms of "t", not "v". Luckily, you have an equation for v in terms of t. v(t) = 25-9.8t. So,

K(t) = 0.5mv^2 = 0.5m[v(t)]^2 = 0.5m(25-9.8t)^2

Functions do not have to have the form "f(x) = something". "K(t) = something" is also a function. "G(duck) = something" is a function. "Hair(lemon, warp) = something" is a function. Does that clear things up?

[phoenirius]

It kind of clears things up but in my book it states that when transforming and merging two formulas into a function that I'm supposed to replace x with either f(x) or g(x). I know what a function means, I'm following the rules that I'm being given and I'm wondering if writing it like that is a function of time.