Two diamonds begin a free fall from rest from the same height, 1.0 s apart. How long after the first diamond begins to fall will the two diamonds be 10 m apart? Take `g = 10 m//s^2.`

Explanation

To solve the problem, we need to find out how long after the first diamond begins to fall the two diamonds will be 10 meters apart. Let's break it down step by step.

### Step 1: Understand the motion of the diamonds
Both diamonds start from rest and fall freely under the influence of gravity. The first diamond starts falling at time \( t = 0 \) seconds, while the second diamond starts falling at \( t = 1 \) second.

### Step 2: Write the equations of motion
The distance fallen by an object in free fall from rest is given by the equation:
\[ h = ut + \frac{1}{2}gt^2 \]
Since both diamonds start from rest, the initial velocity \( u = 0 \). Therefore, the distance fallen by each diamond can be expressed as:

- For the first diamond (falling for time \( t \)):
\[ h_1 = \frac{1}{2}gt^2 \]

- For the second diamond (falling for time \( t - 1 \)):
\[ h_2 = \frac{1}{2}g(t - 1)^2 \]

### Step 3: Set up the equation for the distance apart
We want to find the time when the distance between the two diamonds is 10 meters:
\[ h_1 - h_2 = 10 \]

Substituting the expressions for \( h_1 \) and \( h_2 \):
\[ \frac{1}{2}gt^2 - \frac{1}{2}g(t - 1)^2 = 10 \]

### Step 4: Simplify the equation
Factor out \( \frac{1}{2}g \):
\[ \frac{1}{2}g \left( t^2 - (t - 1)^2 \right) = 10 \]

Now, we can simplify \( (t - 1)^2 \):
\[ (t - 1)^2 = t^2 - 2t + 1 \]
Thus,
\[ t^2 - (t - 1)^2 = t^2 - (t^2 - 2t + 1) = 2t - 1 \]

Substituting back into the equation gives:
\[ \frac{1}{2}g(2t - 1) = 10 \]

### Step 5: Solve for \( t \)
Now, substituting \( g = 10 \, \text{m/s}^2 \):
\[ \frac{1}{2}(10)(2t - 1) = 10 \]
\[ 5(2t - 1) = 10 \]
Dividing both sides by 5:
\[ 2t - 1 = 2 \]
Adding 1 to both sides:
\[ 2t = 3 \]
Dividing by 2:
\[ t = \frac{3}{2} \, \text{s} = 1.5 \, \text{s} \]

### Final Answer
The two diamonds will be 10 meters apart 1.5 seconds after the first diamond begins to fall.

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