To solve the problem, we will follow these steps:
### Step 1: Calculate the time taken to fall from 45 m to 25 m
We can use the second equation of motion to find the time taken to fall from the height of 45 m to 25 m. The distance fallen is:
\[
h = 45 \, \text{m} - 25 \, \text{m} = 20 \, \text{m}
\]
Using the equation:
\[
s = \frac{1}{2} g t^2
\]
where \( g = 10 \, \text{m/s}^2 \), we can rearrange it to find \( t \):
\[
20 = \frac{1}{2} \times 10 \times t^2
\]
\[
20 = 5t^2
\]
\[
t^2 = 4 \implies t = 2 \, \text{s}
\]
### Step 2: Determine the time taken to reach the ground after the explosion
The total time taken to reach the ground from 45 m is calculated as follows:
Using the same equation for the total height:
\[
45 = \frac{1}{2} \times 10 \times t^2
\]
\[
45 = 5t^2
\]
\[
t^2 = 9 \implies t = 3 \, \text{s}
\]
After the explosion, the time taken to fall from 25 m to the ground is:
\[
t_{fall} = 3 \, \text{s} - 2 \, \text{s} = 1 \, \text{s}
\]
### Step 3: Analyze the horizontal motion after the explosion
After the explosion, the two pieces move horizontally. The heavier piece (mass \( 2m \)) moves with a horizontal velocity of \( 10 \, \text{m/s} \). Let the lighter piece (mass \( m \)) have a horizontal velocity \( v_1 \).
Using conservation of momentum in the horizontal direction:
\[
m v_1 - 2m \cdot 10 = 0
\]
This simplifies to:
\[
v_1 = 20 \, \text{m/s}
\]
### Step 4: Calculate the relative velocity of separation
The relative velocity of separation between the two pieces after the explosion is:
\[
v_{relative} = v_1 + v_2 = 20 \, \text{m/s} + 10 \, \text{m/s} = 30 \, \text{m/s}
\]
### Step 5: Calculate the distance between the two pieces when they strike the ground
The time interval after the explosion until they hit the ground is \( 1 \, \text{s} \). Therefore, the distance between them when they strike the ground is:
\[
\text{Distance} = v_{relative} \times \text{time} = 30 \, \text{m/s} \times 1 \, \text{s} = 30 \, \text{m}
\]
### Final Answer
The distance between the two pieces when both of them strike the ground is **30 meters**.
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