The free particle

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The free particle is a particle that is not acted on by a force, i.e. F(x)=0. Then, Newton's second law reduces to

or simply

This needs to be solved subject to the initial conditions, and , where and are constants. Since , we can also write

which tells us that v(t) is a constant for all time, v(t) = C. Applying the initial conditions tells us so , and we have , i.e. a constant for all time. Now, the position, x(t) is given by integrating the velocity

where B is a constant of integration. Since , we have , so that

Note that this is just the equation for a line. If we plot x(t) vs. t, we find

  
Figure 1:

which shows that Newton's first law follows from Newton's second law for the special case that all of the forces are 0.

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