Two wheels with diameters 7 cm and 14 cm start rolling simultaneously from points X and Y, which are 1980 cm apart, towards each other in opposite directions. Both wheels make the same number of revolutions per second and they meet after 10 seconds. What is the linear speed of the smaller wheel in centimetres per second?

Introduction / Context:
This problem combines ideas from circular motion and relative speed. Two wheels roll towards each other and meet after a certain time. The key is that both wheels complete the same number of revolutions per second, so their linear speeds are proportional to their radii or diameters. Using the distance between them and the time until they meet, we can compute the speed of the smaller wheel.

Given Data / Assumptions:

• Diameter of smaller wheel = 7 cm
• Diameter of larger wheel = 14 cm
• Distance between starting points X and Y = 1980 cm
• They roll towards each other and meet after 10 seconds
• Both have the same number of revolutions per second

Concept / Approach:
If two wheels make the same number of revolutions per second, their linear speeds are proportional to their circumferences, and hence to their diameters. Therefore, the larger wheel moves twice as fast as the smaller one because its diameter is twice as large. Let the speed of the smaller wheel be v cm/s. Then the speed of the larger wheel is 2v. Since they approach each other, their relative speed is v + 2v. Using distance = speed * time, we can solve for v.

Step-by-Step Solution:
Let speed of smaller wheel = v cm/s Speed of larger wheel = 2v cm/s (since diameter ratio is 1:2) When moving towards each other, relative speed = v + 2v = 3v cm/s Distance between them = 1980 cm Time = 10 s, so distance = relative speed * time 1980 = 3v * 10 = 30v v = 1980 / 30 = 66 cm/s

Verification / Alternative check:
In 10 seconds, the smaller wheel covers 66 * 10 = 660 cm. The larger wheel covers 132 * 10 = 1320 cm. Total distance covered by both = 660 + 1320 = 1980 cm, which matches the initial separation. This confirms that the smaller wheel speed is 66 cm/s.

Why Other Options Are Wrong:
44 cm/s: Would give total distance 44*10 + 88*10 = 1320 cm only. 88 cm/s and 110 cm/s: These lead to total distances greater than 1980 cm in 10 seconds. 132 cm/s: Implies an excessively high speed and total distance far beyond 1980 cm.

Common Pitfalls:
Forgetting that linear speed is proportional to diameter when revolutions per second are equal. Adding distances incorrectly when using relative speed. Mistakes in division when solving for v from 1980 = 30v.

Final Answer:
The speed of the smaller wheel is 66 cm/s