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**Free fall**

It is quite common that we know the rate of change of something, but not the ting itself. Say for example we know that the acceleration is a constant, *a*. We know that the acceleration is the rate of change of the rate of change of position, i.e that

where s is the distance (position) and the second derivative is the derivative with respect to time twice.

The derivative of the position (the rate of change of the distance), so the antiderivative of the above will give us the velocity. We get

But the constant must be the initial velocity (put *t*=0) so we have

A second antiderivative gives us

Here we could reason that the constant must be the initial position.

**Simple harmonic motion**

Say we have a mass hanging in a spring that obeys Hooke´s law,

The total force will thus be

Here up is in the positive direction and *y*_{0} is the height at which the spring force and gravity balance each other (the equilibrium point).

We have that

or

We thus want to find a function such that its second derivative is (except for a constant factor) equal to the negative of itself. We have that

To make it work we may multiply *t* with a constant, *ω*, called the angular velocity. This gives us

So with

substituted into our original equation we get

So

And thus that

This is a particular solution, and it tells us that the motion of the mass can be like a sine wave. This we have derived from basic physics. The general solution is

where *A* is the amplitude of the motion, and *θ* is the so called phase angle. You can verify that this is a solution yourself by finding the second derivative of the above and verify that it satisfies the original differential equation.

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