Two motorcycles are riding around a circular track at the same angular velocity. One motorcycle is at a radius of 15 m; and the second is at a radius of 18 m. What is the ratio of their linear speeds, v2/v1

v₂ / v₁ = 1.2

Explanation:

By definition the angular velocity is the rate of change of the angle traveled respect from time, as follows:

       

Now by definition of angle, we can replace in (1) Δθ, by the following expression:

       

Replacing (2) in (3)  we have, since :Now, by definition of linear velocity, we know that Δs/Δt = v.Replacing in (3), we have a fixed relationship between angular and linear velocity, as follows:

       

Now, since we know that the angular velocity for both motorcycles is the same, if we call r₁ to the smaller radius (15 m), we can write the following proportion:

       

Rearranging terms, and replacing by the values of the radii, we have:

       

The ratio of their linear speeds, v2/v1, is just the relationship of their radii, i.e., 1.2.

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