How Long Does it Take for a Faster Train to Overtake a Slower One?
Understanding the dynamics of two trains traveling at different speeds is an interesting problem that often appears in mathematics and physics. This article explores the scenario where two trains start moving in the same direction with significant differences in their speeds. Specifically, we will determine how long it takes for a 65 mph train to overtake a 50 mph train, given a 3-hour head start for the slower train. Let's break this down step-by-step and explain the essential components using a detailed analysis.
Defining Variables and Initial Conditions
First, let's define the key variables and parameters:
s1 50 mph - Speed of the first train (Train A) s2 65 mph - Speed of the second train (Train B) t_diff 3 hours - Time difference, which is the head start for Train ANow, let's calculate the distance covered by Train A during the 3-hour head start:
d1 s1 * t_diff 50 mph * 3 hours 150 miles
Relative Speed Calculation
To determine how long it takes for Train B to overtake Train A, we need to understand the relative speed difference. This is the key factor in determining the time taken for the overtake:
relative_speed s2 - s1 65 mph - 50 mph 15 mph
The relative speed of 15 mph represents how much faster Train B is moving compared to Train A.
Calculating the Overtake Time
We use the distance formula to find the time it takes for Train B to cover the 150-mile gap between the two trains:
time_to_overtake distance / relative_speed 150 miles / 15 mph 10 hours
Therefore, it takes Train B 10 hours to overtake Train A. During this time, Train B has traveled 650 miles (65 mph * 10 hours), while Train A has traveled 550 miles (50 mph * 13 hours 50 mph * (10 3) hours), covering the initial 150-mile gap plus an additional 400 miles.
Verification by Calculation
We can also verify this result by another method. Let's assume Train B takes t hours to catch up with Train A:
s2 * t s1 * (t t_diff)
Substituting the known values:
65 * t 50 * (t 3)
Expanding and solving for t:
65t 50t 150
15t 150
t 150 / 15 10 hours
This confirms that it indeed takes 10 hours for Train B to overtake Train A.
Conclusion
In conclusion, it takes 10 hours for the 65 mph train to overtake the 50 mph train that had a 3-hour head start. This solution is verified through both direct calculation and an algebraic approach, ensuring the accuracy of our result.