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

v₂ / v₁ = 1.2

Step-by-step explanation:

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

`\omega = (\Delta \theta)/(\Delta t) (1)`

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

`\Delta \theta = (\Delta s)/(r) (2)`

• Replacing (2) in (3) we have, since :
• 
  `\omega = (\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 =(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:

`(v_(1) )/(r_(1) ) = (v_(2))/(r_(2)) (5)`

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

`(v_(2) )/(v_(1)) =(r_(2) )/(r_(1) ) =(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.