1. The problem statement, all variables and given/known data I am trying to understand game of roulette through kinematic equations of circular motion. Roulette is game where a ball spins in circular motion. Initially it is accelerated such that velocity of ball increases with time however after reaching its peak value it starts to decelerate until it finally leaves its circular track and falls in rotar. (ref: attached image) Spoiler Given: Initial angular displacement of ball at start θ0=0 rad Initial angular velocity of ball at start ω0=0 rad/s Angular displacement of ball in one revolution θ1= 2∏ rad = 6.28318548 rad Angular displacement of ball in initial six revolutions θ6= 6 * θ1= 37.6991 rad Total angular displacement of ball before it leaves the outer track θtot=114.1162 rad (i.e. slightly more than 18 revolutions) Angular displacement of ball after its initial six revolutions before it leaves the outer track θf= θtot-θ6=76.4171 rad Time taken by ball to complete initial six revolutions t6= 3.875 s Total time taken by ball before it leaves the outer track ttot=22.479 s Time taken by ball after its initial six revolutions before it leaves the outer track tf=ttot-t6=18.604 s 2. Relevant equations ω2=ω1+σt ---------- eq:(1) θ=ω1t+(1/2σt2) ---------- eq:(2) ω22=ω12+2σθ ---------- eq:(3) where, ω1 and ω2 are initial and final angular velocities respectively in rad/s σ is angular acceleration in rad/s2 θ is angular displacement in radians t is time in seconds 3. The attempt at a solution Angular velocity of ball at sixth revolution can be calculated as follows: ω6=θ6/t6=9.7288 rad/s Similarly, Angular velocity of ball before it leaves the outer track can be calculated as follows: ωf=θf/tf=4.10 rad/s Since ω6>ωf, angular acceleration σ must be negative.. Using eq:(1) σ=(ωf-ω6)/t Here, t would be time taken by ball to reach ωf from ω6 i.e. ttot-t6=tf ∴σ=(ωf-ω6)/tf ∴σ=-0.3025 rad/s2 However if we use eq:(3) σ=(ωf2-ω62)/2θ Here, θ would be angular displacement of ball to reach ωf from ω6 = θtot-θ6=θf ∴σ=(ωf2-ω62)/2θf ∴σ=-0.5093 rad/s2 My question! How do i get two different values of angular deceleration constant σ? The values in "Given:" section are real time observations still why there is difference in σ when it is calculated by different equations?