Linear motion under a variable force
Variable force
- When the force depends on position $x$, write acceleration as:
$$a = v\frac{dv}{dx}$$
- This turns Newton's law $F = ma$ into a differential equation linking $v$ and $x$, which you separate and integrate.
Practice
When a force depends on position x, acceleration is best written as:
a = v dv/dx lets you relate velocity to position directly.
Practice
Using a = v dv/dx turns Newton's law into:
F = ma becomes a separable differential equation in v and x.
Practice
You then separate the variables and integrate to find v in terms of x.
Separating and integrating both sides gives the velocity as a function of position.
You've got it
Key idea
- for a position-dependent force, use $a = v\dfrac{dv}{dx}$
- this makes $F = ma$ a differential equation in $v$ and $x$
- separate and integrate to find $v$ as a function of $x$