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

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cjrodriguez89 cjrodriguez89 · Answer 1 · 2020-11-03T02:31:14+01:00

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:

[tex]\omega = \frac{\Delta \theta}{\Delta t} (1)[/tex]

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

[tex]\Delta \theta = \frac{\Delta s}{r} (2)[/tex]

Replacing (2) in (3) we have, since :
[tex]\omega = \frac{\Delta s}{\Delta t*r} (3)[/tex]
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:

[tex]\omega =\frac{v}{r} (4)[/tex]

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:

[tex]\frac{v_{1} }{r_{1} } = \frac{v_{2}}{r_{2}} (5)[/tex]

[tex]\frac{v_{2} }{v_{1}} =\frac{r_{2} }{r_{1} } =\frac{18 m}{15 m} = 1.2[/tex]

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