Linear motion under a variable force
Introduced| English | Meaning |
|---|---|
| drag/dræɡ/ | A resistive force exerted by a fluid on an object moving relative to it. |
| position/pəˈzɪʃn/ | An object's location relative to a chosen reference point or coordinate system. |
| chain rule/tʃeɪn ruːl/ | The rule for differentiating a composite function by multiplying the appropriate derivatives. |
| separable/ˈsepərəbl/ | Describes a differential equation that can be rearranged so that each variable and its differential appear on separate sides. |
| separate/ˈseprət/ | separate |
| integrate/ˈɪntɪɡreɪt/ | To find an indefinite or definite integral of a function. |
| boundary condition/ˈbaʊndəri kənˈdɪʃn/ | A specified value or relationship that a solution must satisfy at a boundary or stated position, helping determine constants or allowable solutions. |
When the force keeps changing
- Gravity near Earth is roughly constant — but the pull of a planet, the drag 阻力 on a car, or a magnetic force all change with position 位置.
- A force that depends on where you are needs a cleverer form of Newton's law.
Acceleration as a function of position
- When the force (and so the acceleration) depends on position $x$, rewrite acceleration as:
- This comes from the chain rule 链式法则: $\dfrac{dv}{dt} = \dfrac{dv}{dx}\dfrac{dx}{dt} = v\dfrac{dv}{dx}$.
When force varies with position, the work done is the area under the F–x graph
Work from a variable force
W = ∫ F dx
When the force changes, the work done is the area under the force–distance graph.
When a force depends on position x, acceleration is best written as:
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a = dv/dt = (dv/dx)(dx/dt) = v dv/dx — convenient when force depends on x.
The identity a = v dv/dx comes from the ______ rule.
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dv/dt = (dv/dx)(dx/dt) = v dv/dx by the chain rule.
Turning F = ma into a differential equation
- Substituting gives $F(x) = mv\dfrac{dv}{dx}$ — a separable 可分离 differential equation linking $v$ and $x$.
- Separate 分离 the variables ($v\,dv$ on one side, $x$ on the other) and integrate 积分.
When to use which form. Force depends on time → use $a = \dfrac{dv}{dt}$. Force depends on position → use $a = v\dfrac{dv}{dx}$. Choosing the right one is half the battle.
Substituting a = v dv/dx into F = ma gives a separable differential equation in v and x.
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F(x) = mv dv/dx separates into v dv and an x-expression, then integrate.
If instead the force depends on time t, the natural form for acceleration is:
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Time-dependent force → a = dv/dt; position-dependent → a = v dv/dx.
The method
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- Write $F = ma$ with $a = v\dfrac{dv}{dx}$.
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- Separate: gather $v$ terms with $dv$, $x$ terms with $dx$.
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- Integrate both sides, then apply the boundary conditions 边界条件 to find $v$ as a function of $x$.
- When the force depends on position, treat it as a variable force with $a=v\,dv/dx$.
Put the steps for solving motion under a position-dependent force in order.
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Set up with v dv/dx, separate, integrate, then use the conditions to fix the constant.
You've got it
- for a position-dependent force, use $a = v\dfrac{dv}{dx}$ (from the chain rule)
- this makes $F = ma$ a separable differential equation in $v$ and $x$
- separate and integrate to find $v$ as a function of $x$