Linear motion under a variable force

Need to know

What you need to know

• **If force depends on time, $F(t)$:** Use $a = \frac{dv}{dt}$. The equation of motion is $F(t) = m\frac{dv}{dt}$. You can integrate with respect to $t$ to find $v(t)$.
• **If force depends on displacement, $F(x)$:** Use $a = v\frac{dv}{dx}$. The equation of motion is $F(x) = mv\frac{dv}{dx}$. You can solve by separating variables to find a relationship between $v$ and $x$.
• **If force depends on velocity, $F(v)$:** You have a choice. Use $a = \frac{dv}{dt}$ to find $v$ as a function of $t$, or use $a = v\frac{dv}{dx}$ to find $v$ as a function of $x$. The question will guide you.

Explanation

Motion on the Move

1. Identify the variable force $F$ and use Newton's Second Law ($F=ma$) to create an equation of motion.
2. Choose the correct form of acceleration: $\frac{dv}{dt}$ if force depends on time, or $v\frac{dv}{dx}$ if force depends on displacement.
3. Substitute the expression for acceleration into your equation to form a first-order differential equation.
4. Solve the differential equation by separating variables and integrating, using given conditions to find the constant of integration.