To solve this problem, we need to determine how long it will take the second cyclist, traveling at a faster speed, to catch up to the first cyclist.

Given Data:

1. Speed of the first cyclist: $v_1 = 19 \, \text{km/h}$
2. Speed of the second cyclist: $v_2 = 30 \, \text{km/h}$
3. Distance traveled by the first cyclist before the second cyclist starts: $d = 24.2 \, \text{km}$

Key Idea:

The second cyclist will catch up to the first when they have traveled the same distance after the second cyclist starts. This is equivalent to solving for the time $t$ at which the second cyclist makes up the 24.2 km head start of the first cyclist.

Steps to Solve:

1. Relative Speed:
   The second cyclist is traveling faster than the first, so the relative speed between the two is:
   $v_{\text{relative}} = v_2 - v_1 = 30 - 19 = 11 \, \text{km/h}.$

2. Time to Catch Up:
   Using the formula $\text{time} = \frac{\text{distance}}{\text{speed}}$, we calculate the time it takes for the second cyclist to close the 24.2 km gap:
   $t = \frac{d}{v_{\text{relative}}} = \frac{24.2}{11}.$

3. Calculation:
   $t \approx 2.2 \, \text{hours}.$

Final Answer:

It will take approximately 2.2 hours (or 2 hours and 12 minutes) for the second cyclist to catch up to the first.

Do you want further clarification on any step? Here are five related questions for deeper exploration:

1. What would happen if the second cyclist's speed was less than the first cyclist's speed?
2. How can we convert this result into minutes or seconds for more precision?
3. What if the initial gap was different, say 50 km instead of 24.2 km?
4. How can this problem be visualized on a graph showing their positions over time?
5. How would the solution change if both cyclists had different starting times?

Tip: For problems involving relative motion, always focus on the difference in speeds and initial gaps. This simplifies calculations.