Question

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

Answer 1

Answer:

v₂ / v₁ = 1.2

Explanation:

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

        $\omega = \frac{\Delta \theta}{\Delta t} (1)$

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

       $\Delta \theta = \frac{\Delta s}{r} (2)$

• Replacing (2) in (3)  we have, since :
• $\omega = \frac{\Delta s}{\Delta t*r} (3)$
• 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:

       $\omega =\frac{v}{r} (4)$

• 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:

        $\frac{v_{1} }{r_{1} } = \frac{v_{2}}{r_{2}} (5)$

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

       $\frac{v_{2} }{v_{1}} =\frac{r_{2} }{r_{1} } =\frac{18 m}{15 m} = 1.2$

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