Quick Recap — Advanced Kinematics (Multi-step)

  • Combine equations: eliminate time or use v2=u2+2asv^2=u^2+2as for speed–distance; for two bodies write each one's x(t)x(t) and set them equal (meeting) or subtract (separation).
  • Projectile from a height / at an angle: treat the vertical motion with initial vertical velocity usin⁡θu\sin\theta and drop hh; solve the quadratic for total time.
  • Relative velocity 2D: to cross a river in minimum time, head straight across (drift =vriver×t=v_{river}\times t); for shortest path, angle upstream so the resultant is perpendicular.
  • Circular: change in velocity over a turn of angle ϕ\phi has magnitude 2vsin⁡ϕ22v\sin\dfrac{\phi}{2}.

Worked mini-example. For u=50u=50 m/s at 37∘37^\circ: R=240R=240 m, H=45H=45 m, T=6T=6 s. Its speed at the top is ucos⁡37∘=50×0.8=40u\cos37^\circ=50\times0.8=40 m/s.