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, `v_(2)//v_(1)`?

Options

A. 1
B. 1.2
C. 0.83
D. 0.71

Answer

To solve the problem of finding the ratio of the linear speeds of two motorcycles riding around a circular track at the same angular velocity, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Angular Velocity and Linear Speed**:
The linear speed \( v \) of an object moving in a circle is related to its angular velocity \( \omega \) and the radius \( r \) of the circle by the formula:
\[ v = \omega r \]

2. **Identify the Given Values**:
- For Motorcycle 1 (at radius \( r_1 \)):
- \( r_1 = 15 \, \text{m} \)
- For Motorcycle 2 (at radius \( r_2 \)):
- \( r_2 = 18 \, \text{m} \)
- Both motorcycles have the same angular velocity \( \omega \).

3. **Express the Linear Speeds**:
- The linear speed of Motorcycle 1:
\[ v_1 = \omega r_1 = \omega \times 15 \]
- The linear speed of Motorcycle 2:
\[ v_2 = \omega r_2 = \omega \times 18 \]

4. **Set Up the Ratio of Linear Speeds**:
To find the ratio of their linear speeds \( \frac{v_2}{v_1} \):
\[ \frac{v_2}{v_1} = \frac{\omega r_2}{\omega r_1} = \frac{r_2}{r_1} \]

5. **Substitute the Values of the Radii**:
\[ \frac{v_2}{v_1} = \frac{18}{15} \]

6. **Simplify the Ratio**:
\[ \frac{v_2}{v_1} = \frac{18 \div 3}{15 \div 3} = \frac{6}{5} \]

7. **Convert to Decimal**:
\[ \frac{6}{5} = 1.2 \]

### Final Answer:
The ratio of their linear speeds \( \frac{v_2}{v_1} \) is \( 1.2 \).

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