A train leaves Atlanta at 1:00 PM. A second train leaves the same city in the same direction at 5:00 PM. The second train travels 48 mph faster than the first. If the second train overtakes the first at 9:00 PM, what is the speed of each of the two trains?

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To solve this problem, let's break it down step by step:

Step 1: Calculate the time it takes for the second train to overtake the first train.
The first train leaves at 1:00 PM and the second train leaves at 5:00 PM, so the second train has a 4-hour head start.

Step 2: Determine the distance between the two trains.
Since the second train overtakes the first train, they must meet at the same point. Therefore, the distance traveled by both trains will be the same.

Step 3: Set up equations for the distance traveled by each train.
Let's assume the speed of the first train is "x" mph. The speed of the second train is 48 mph faster than the first train, so it will be "x + 48" mph.

Distance = Speed * Time (d = s * t)

For the first train:
Distance1 = x * (t1 + 4) (t1 is the time taken by the first train to get overtaken.)

For the second train:
Distance2 = (x + 48) * t1

Step 4: Equate the distances traveled by both trains.
Since the distance is the same, we can set up an equation:
x * (t1 + 4) = (x + 48) * t1

Step 5: Solve the equation for x (speed of the first train).
Expand the equation:
xt1 + 4x = xt1 + 48t1

Simplify:
4x = 48t1

Divide both sides by 4:
x = 12t1

Step 6: Calculate the time t1.
We know that the second train overtakes the first at 9:00 PM, which is 8 hours after the first train leaves (1:00 PM to 9:00 PM).

t1 + 4 = 8

Subtract 4 from both sides:
t1 = 4

Step 7: Calculate the speed of the first train (x).
x = 12t1
x = 12 * 4
x = 48 mph

Step 8: Calculate the speed of the second train (x + 48).
Speed of the second train = x + 48
= 48 + 48
= 96 mph

Therefore, the speed of the first train is 48 mph, and the speed of the second train is 96 mph.