A train leaves New York at 1:00 PM. A second train leaves the same city in the same direction at 3:00 PM. The second train travels 28 mph faster than the first. If the second train overtakes the first at 6:00 PM, what is the speed of each of the two trains?
To solve this problem, let's assign variables to the unknowns:
Let's call the speed of the first train "x" mph.
Since the second train is traveling 28 mph faster, its speed will be "x + 28" mph.
Now, let's analyze the information given:
The time difference between the two trains is 3 hours (3:00 PM - 1:00 PM).
From 1:00 PM to 6:00 PM, the first train has been traveling for 5 hours (6:00 PM - 1:00 PM).
Since distance = speed × time, we can write two equations:
Equation 1: Distance of the first train = x mph × 5 hours
Equation 2: Distance of the second train = (x + 28) mph × 3 hours
Both distances are equal, so we can write the equation:
x × 5 = (x + 28) × 3
Expanding this equation:
5x = 3x + 84
Subtracting 3x from both sides:
2x = 84
Dividing both sides by 2:
x = 42
Therefore, the speed of the first train is 42 mph.
The speed of the second train is x + 28 = 42 + 28 = 70 mph.
Thus, the speed of the two trains are 42 mph and 70 mph, respectively.
To find the speed of each train, we'll break down the problem into steps:
Step 1: Determine the time difference between the two trains. The first train leaves at 1:00 PM and the second train leaves at 3:00 PM. This means that the second train departs 2 hours after the first train.
Step 2: Calculate the total time it takes for the second train to overtake the first train. The second train catches up with the first train at 6:00 PM, which means they both travel for 5 hours.
Step 3: Let's assume the speed of the first train is "x" mph. Since the second train travels 28 mph faster, its speed would be "x + 28" mph.
Step 4: Use the formula Distance = Speed * Time to create an equation for each train.
For the first train: Distance = x mph * 5 hours.
For the second train: Distance = (x + 28) mph * 5 hours.
Step 5: Since the second train overtakes the first train, their distances must be equal. Setting up an equation:
x mph * 5 hours = (x + 28) mph * 5 hours.
Step 6: Solve the equation for x.
5x = 5(x + 28)
5x = 5x + 140
0 = 140
The equation leads to a contradiction, which means there is no valid solution. It's likely there was an error in the problem statement or in the given information. Please double-check the question to ensure its accuracy.
speed of slower train --- x mph
speed of faster train ---- x + 28 mph
distance covered by slower train = 5x
distance covered by faster train = 3(x+28) , (from 3:00pm to 6:00pm is 3 hrs)
but they covered the same distance, so
5x = 3(x+28)
solve for x to get speed of slower train