We cannot solve equation {eq}`generalEOM` for a general function $F$. However, in each of the special cases that the force depends on only one of the three variables, we can find a general solution, albeit as an integral over the force, which we may or may not be able to calculate explicitly.
We cannot solve equation {eq}`generalEOM` for a general function $F$. However, in each of the special cases that the force depends on only one of the three variables, we can find a general solution, albeit as an integral over the force, which we may or may not be able to calculate explicitly.
### Case 1: F = F(t)
### Case 1: $F = F(t)$
If the force only depends on time, we can solve equation {eq}`generalEOM` by direct integration. Using that $v = \dot{x}$, we have $m \dot{v} = F(t)$, which we integrate to find
If the force only depends on time, we can solve equation {eq}`generalEOM` by direct integration. Using that $v = \dot{x}$, we have $m \dot{v} = F(t)$, which we integrate to find
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x(t) = \int_{t_0}^t v(t') \mathrm{d}t'.
x(t) = \int_{t_0}^t v(t') \mathrm{d}t'.
$$ (positionvelocityintegral)
$$ (positionvelocityintegral)
### Case 2: F = F(x)
### Case 2: $F = F(x)$
If the force depends only on the position in space (as is the case for the harmonic oscillator), we cannot integrate over time, as to do so we would already need to know $x(t)$. Instead, we invoke the chain rule to rewrite our differential equation as an equation in which the position is our variable. We have:
If the force depends only on the position in space (as is the case for the harmonic oscillator), we cannot integrate over time, as to do so we would already need to know $x(t)$. Instead, we invoke the chain rule to rewrite our differential equation as an equation in which the position is our variable. We have:
which gives us $t(x)$. In principle we can invert this expression to give us $x(t)$, although in practice this may not be easy.
which gives us $t(x)$. In principle we can invert this expression to give us $x(t)$, although in practice this may not be easy.
### Case 3: F = F(v)
### Case 3: $F = F(v)$
If the force depends only on the velocity, there are two ways we can proceed. We can write the equation of motion as $m \mathrm{d}v / \mathrm{d} t = F(v)$ and use separation of variables to get:
If the force depends only on the velocity, there are two ways we can proceed. We can write the equation of motion as $m \mathrm{d}v / \mathrm{d} t = F(v)$ and use separation of variables to get:
"question":"An 80-ton airplane stands on the tarmac. The airplane is 50 meters long; the main wheels are 27 meters from the nose, and the nose wheel is 3 meters from the nose. <br /> How large is the force on the nose wheel?<br />(We assume the mass of the plane is distributed uniformly.)",
"question":"An 80-ton airplane stands on the tarmac. The airplane is 50 meters long; the main wheels are 27 meters from the nose, and the nose wheel is 3 meters from the nose. How large is the force on the nose wheel?<br />(We assume the mass of the plane is distributed uniformly.)",
