Orbit & Curve Lab
Differentiate position vectors, control circular motion and navigate a current.
Motion in two dimensions
Velocity is tangent to the path. Acceleration describes changes in speed or direction.
Speed history
The physics
Vector position: r(t) = x(t)i + y(t)j, in metres.
Velocity: v(t) = dr/dt. Its components are dx/dt and dy/dt, in m/s.
Acceleration: a(t) = dv/dt, in m/s².
Speed: |v| = √(vₓ² + vᵧ²), in m/s.
Circular motion: normal acceleration magnitude aₙ = v²/r, towards the centre.
Non-uniform circular motion: total acceleration magnitude |a| = √(aₜ² + aₙ²), because tangential and normal acceleration are perpendicular.
Angular motion: ω = v/r. With constant tangential acceleration aₜ, angular acceleration α = aₜ/r and θ(t) = θ₀ + ω₀t + ½αt².
Relative velocity: v_ground = v_boat/water + v_water/ground, added as vectors.
River crossing: take east as +x and the far bank as +y. With heading θ measured eastward from north, vₓ = U sinθ + current and vᵧ = U cosθ.
Crossing time: T = river width / vᵧ for positive vᵧ.
All examples use idealised point objects. Circular trajectories have fixed radius; river current and boat speed relative to water are constant.
Six university mechanics challenges completed.