State-Dependent Forces and Differential Dynamics
In one dimension a=dv/dt=v dv/dx; insert the useful form into m a=F(x,v,t).
Why this shows up in the exam
v as a function of x · Linear or quadratic drag · Position-dependent force
Learn the idea
Match the acceleration derivative to the variable used by the force or motion data. Position- or speed-dependent forces make constant-acceleration formulas unsafe.
🧠 Memory hook: Use the derivative that matches the data.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- a = v dv/dx — velocity known versus position
- m dv/dt = mg-kv — linear downward drag
- v_terminal = mg/k — linear-drag terminal speed
How to approach it
- 1Choose signed coordinate
- 2Rewrite acceleration suitably
- 3Integrate and check limiting behavior
Common slip-ups that cost marks
- •Constant-acceleration formulas
- •Wrong drag sign
- •Linear formula for quadratic drag
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A 2 kg block moves on a horizontal rough surface with coefficient of kinetic friction 0.2. A horizontal force of 10 N acts on it. Take g = 10 m/s^2. What is its acceleration?
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