Elastic Collisions in 2D (Edexcel A-Level Further Mathematics): Flashcards

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Successive Collisions with a Surface
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Parallel component in oblique collision (smooth surface)

Remains unchanged

Formula: perpendicular velocity after collision

v,after=ev,beforev_{\perp, \text{after}} = -e v_{\perp, \text{before}}

Range of coefficient of restitution ee

0e10 \leq e \leq 1

Rebound height after nnth collision

hn=e2nh0h_n = e^{2n} h_0

Perpendicular velocity after nn collisions

v,n=(e)nv,0v_{\perp, n} = (-e)^n v_{\perp, 0}

Why parallel component unchanged?

No impulse in tangential direction (smooth surface)

Vector decomposition of velocity v\mathbf{v}

v=v+v\mathbf{v} = \mathbf{v}_\parallel + \mathbf{v}_\perp

Speed after collision (rebound)

vrebound=evimpactv_\text{rebound} = e v_\text{impact}

Loss of kinetic energy formula

ΔKE=KEinitialKEfinal\Delta KE = KE_\text{initial} - KE_\text{final}

Perpendicular velocity after impact (simplified)

v=euv_\perp = -e u_\perp

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