A train traveling 50 mph left a station 30 minutes before a second train running at 55 mph. How soon did the second train overtake the first?
If 50x represents the distance the slower train travels, then the faster train travels
55(x + 0.5)
50 + 30x
55(x - 0.5)
We can set up a distance equation based on the information given.
Let t be the amount of time it takes for the second train to catch up to the first train in hours.
The distance the slower train traveled is given by 50x, where x is the time it takes for the slower train to travel.
The distance the faster train traveled is given by 55(x - 0.5), where x - 0.5 represents the time it takes for the faster train to travel, since it left 30 minutes after the slower train.
Since the two trains have traveled the same distance when they meet, we can set up the equation:
50x = 55(x - 0.5)
Now, we can solve for x:
50x = 55x - 27.5
-5x = -27.5
x = 5.5
Since x represents the time it takes for the slower train to travel, it took 5.5 hours for the slower train to travel.
The second train caught up to the first train 30 minutes after it left the station, so the total time it took for the second train to catch up to the first train is 5.5 + 0.5 = 6 hours.
Therefore, the second train overtook the first train 6 hours after the first train left the station.
Let's solve the problem step by step:
1. Let's assume that the second train overtakes the first train after a time of 't' hours since the second train started.
2. Since the first train has a head start of 30 minutes, it has already traveled a distance of (50 mph * 0.5 hours) = 25 miles.
3. The speed at which the second train is traveling relative to the first train is the difference between their speeds: (55 mph - 50 mph) = 5 mph.
4. Therefore, for the second train to catch up to the first train, it needs to make up a distance of 25 miles, but it is traveling 5 mph faster.
5. So, the time it takes for the second train to catch up can be calculated using the formula: time = distance / relative speed.
6. Plugging in the values, we get: t = 25 miles / 5 mph = 5 hours.
Therefore, the second train overtakes the first train after 5 hours.
To find out how soon the second train overtakes the first train, we need to find the time it takes for both trains to travel the same distance.
Let's assume that the time it takes for the second train to catch up with the first train is represented by 't' hours.
For the first train, we can calculate the distance it travels using the formula:
Distance = Speed × Time
The speed of the first train is 50 mph, and the time it travels is (t + 0.5) hours (since it left 30 minutes = 0.5 hours earlier).
So, the distance traveled by the first train can be represented as:
Distance1 = 50 × (t + 0.5)
Similarly, for the second train, the distance it travels can be calculated using the formula:
Distance = Speed × Time
The speed of the second train is 55 mph, and the time it travels is 't' hours.
So, the distance traveled by the second train can be represented as:
Distance2 = 55 × t
Now, we need to find the time 't' when both distances are equal. Therefore, we can set up the equation:
Distance1 = Distance2
50 × (t + 0.5) = 55 × t
Simplifying the equation:
50t + 25 = 55t
25 = 5t
t = 5
Therefore, the second train overtakes the first train after 5 hours.