Tangential-Normal Acceleration and Curved Paths
For instantaneous speed v and radius of curvature rho, a = a_t t_hat + a_n n_hat, with a_t = dv/dt and a_n = v squared over rho.
Why this shows up in the exam
Nonuniform circular motion · Road and track curvature design · Deriving motion along a specified plane curve
Learn the idea
Curved motion separates acceleration into speed-changing tangential and direction-changing normal parts. A particle can accelerate by speeding up along the tangent, bending toward the centre of curvature, or doing both. The local curvature, not a globally circular path, controls the normal part.
🧠 Memory hook: Tangent changes speed; normal changes direction.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- a_vector = (dv/dt)t_hat + (v²/rho)n_hat — tangential-normal decomposition
- a_t = dv/dt — rate of change of speed
- a_n = v²/rho — inward acceleration caused by curvature
How to approach it
- 1Find speed as a function of time or position
- 2Identify tangent and local normal
- 3Combine perpendicular acceleration components
Common slip-ups that cost marks
- •Using radius from an arbitrary origin as curvature radius
- •Assuming acceleration is perpendicular to velocity
- •Forgetting the tangential component when speed changes
🌟 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 particle has initial speed 2 m/s and constant acceleration 3 m/s^2 for 4 s. What distance does it cover?
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