Given information:
- Two spaceships approach Earth with equal speeds as measured by an observer on Earth.
- The spaceships are approaching from opposite directions.
- A meter stick on one spaceship is measured to be 60 cm long by an occupant of the other ship.

To find:
The speed of each spaceship as measured from Earth.

Explanation:
To solve this problem, we need to use the concept of time dilation from special relativity. According to special relativity, time dilation occurs when an observer moves relative to another observer at a significant fraction of the speed of light.

Step 1: Establish the frame of reference:
- Let's assume that the observer on Earth is at rest and is measuring the speed of the spaceships.
- We will use the observer on Earth as our frame of reference.

Step 2: Understand the concept of time dilation:
- Time dilation occurs when an object is moving relative to an observer at a significant fraction of the speed of light.
- In this case, the spaceships are approaching Earth with equal speeds, so we can assume that they are moving at the same speed relative to Earth.
- Therefore, the time dilation effect will be the same for both spaceships.

Step 3: Apply the time dilation formula:
- The time dilation formula is given by:
Δt' = Δt / √(1 - v^2/c^2)
where:
Δt' = time measured by the moving observer
Δt = time measured by the stationary observer (on Earth)
v = relative velocity between the observer and the moving object
c = speed of light

Step 4: Use the given information to find the speed of the spaceships:
- According to the question, a meter stick on one spaceship is measured to be 60 cm long by an occupant of the other ship.
- Let's assume that the spaceship with the meter stick is spaceship A, and the other spaceship is spaceship B.
- The length contraction formula is given by:
L' = L / √(1 - v^2/c^2)
where:
L' = length measured by the moving observer
L = length measured by the stationary observer (on Earth)
v = relative velocity between the observer and the moving object
c = speed of light

- In this case, the length measured by the occupant of spaceship B (L') is 60 cm.
- The length measured by the observer on Earth (L) is the actual length of the meter stick, which is 100 cm (1 meter).
- Substituting these values into the length contraction formula, we get:
60 = 100 / √(1 - v^2/c^2)

Step 5: Solve for the speed of the spaceships:
- Rearranging the equation, we have:
√(1 - v^2/c^2) = 100/60
√(1 - v^2/c^2) = 5/3

- Squaring both sides of the equation, we get:
1 - v^2/c^2 = 25/9

- Rearranging the equation, we have:
v^2/c^2 =