Therefore, (a) it takes Mike **approximately 13.4 seconds** to ride 30 m, and (b) the unicycle's linear velocity at 30 m is **approximately 8.93 m/s**.
Here's how to solve this problem:
**a) Time taken to ride 30m:**
1. **Convert wheel diameter to meters:** 20 inches * 0.0254 meters/inch = 0.508 meters
2. **Relate distance, wheel diameter, and angular displacement:** Distance = Angular displacement * Wheel diameter. So, Angular displacement = Distance / Wheel diameter = 30 m / 0.508 m = 58.86 radians.
3. **Relate angular displacement, initial angular velocity, angular acceleration, and time:** We can use the equation θ = ω_i * t + 0.5 * α * t^2, where θ is the angular displacement, ω_i is the initial angular velocity, α is the angular acceleration, and t is the time. Solving for t, we get t = (2 * θ) / (ω_i + sqrt(ω_i^2 + 4 * α * θ)) ≈ 13.4 seconds.
**b) Unicycle's linear velocity at 30m:**
1. **Relate linear velocity and angular velocity:** Linear velocity (v) = Wheel diameter * Angular velocity (ω).
2. **Calculate final angular velocity using the equation:** ω_f = ω_i + α * t = 0.75 rad/s + 1.2 rad/s^2 * 13.4 s ≈ 17.54 rad/s.
3. **Calculate linear velocity:** v = 0.508 m * 17.54 rad/s ≈ 8.93 m/s.
Therefore, (a) it takes Mike **approximately 13.4 seconds** to ride 30 m, and (b) the unicycle's linear velocity at 30 m is **approximately 8.93 m/s**.
The probable question can be: Mike is riding his unicycle of wheel diameter 20 in. Mike rides for 30 m with an initial angular velocity of 0.75 rad/s and an acceleration of 1.2 rad/s2. (a) How long does it take to ride the 30 m? (b) What is the unicycles linear velocity at 30 m?