Position, Velocity, and Acceleration Vectors
For a differentiable position r(t), v = dr/dt and a = dv/dt = d^2r/dt^2. If velocity is given as a spatial field, acceleration along a trajectory is the material derivative dv/dt.
Why this shows up in the exam
Reading motion from parametric coordinates · Finding speed and direction at an instant · Analyzing flow-like position-dependent velocity fields
Learn the idea
Velocity and acceleration are successive time derivatives of the position vector. A moving point traces a path, but its instantaneous velocity is tangent to that path and its acceleration describes how the velocity arrow changes in magnitude or direction.
🧠 Memory hook: Differentiate position once for velocity and twice for acceleration.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- v_vector = dr_vector/dt — instantaneous velocity
- a_vector = dv_vector/dt = d²r_vector/dt² — instantaneous acceleration
- a = partial(v)/partial(t) + (v dot grad)v — acceleration for a velocity field
How to approach it
- 1Differentiate each component independently
- 2Substitute the requested time or position
- 3Form magnitude and direction only at the end
Common slip-ups that cost marks
- •Differentiating only the magnitude
- •Ignoring time-dependent unit components
- •Using v/t instead of dv/dt
🌟 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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