Time from a, r, and ω [Ex. 1]
A car decelerates at 6.6 m/s². The tires have a radius of 0.26 m and have a positive direction during the forward motion. If the initial angular velocity is 94 rad/s, how long does it take the car to fully stop?
-- Given values: -> a = 6.6 m/s² -> r = 0.26 m -> ωi = 94 rad/s -> ωf = 0 rad/s
-- Use one of the kinematic equations 1. ωf = ωi + αt 2. Δθ = ωit + (1/2)αt² 3. ωf² = ωi² + 2αΔθ
-- Use Equation 1
-- Missing α, which is angular acceleration
-- Solve for α
α = a/r α = 6.6 / 0.26 α = 25.385 rad/s²
-- Should be negative, because the car is decelerating
α = -25.385 rad/s²
-- Now solve for t using Equation 1
ωf = ωi + α 0 = 94 + (-25.385)t 0 = 94 - 25.385t -94 = -25.385t 94 = 25.385t 94 / 25.385 = t 3.7 = t
-- It takes the car 3.7 seconds to stop
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