Two bodies begin a free fall from rest from the same height 2 seconds apart. How long after the first body begins to fall, the two bodies will be 40 m apart? `("Take g = 10ms"^(-2))`

Options

A. 1 s
B. 2 s
C. 3 s
D. 4 s

Answer

To solve the problem of two bodies falling from rest, we will break it down step by step.

### Step 1: Understanding the problem
We have two bodies falling from the same height:
- Body 1 starts falling at time \( t = 0 \) seconds.
- Body 2 starts falling 2 seconds later, at \( t = 2 \) seconds.

We need to find out how long after Body 1 starts falling, the two bodies will be 40 meters apart.

### Step 2: Equations of motion
For an object in free fall, the distance \( s \) fallen after time \( t \) can be calculated using the equation:
\[ s = ut + \frac{1}{2}gt^2 \]
where:
- \( u \) is the initial velocity (which is 0 for both bodies),
- \( g \) is the acceleration due to gravity (given as \( 10 \, \text{m/s}^2 \)),
- \( t \) is the time in seconds.

### Step 3: Distance fallen by Body 1
For Body 1, which falls for time \( t \):
\[ s_1 = 0 + \frac{1}{2} \cdot 10 \cdot t^2 = 5t^2 \]

### Step 4: Distance fallen by Body 2
Body 2 starts falling at \( t = 2 \) seconds. Therefore, when Body 1 has fallen for \( t \) seconds, Body 2 has been falling for \( t - 2 \) seconds (only valid when \( t > 2 \)):
\[ s_2 = 0 + \frac{1}{2} \cdot 10 \cdot (t - 2)^2 = 5(t - 2)^2 \]

### Step 5: Finding the distance between the two bodies
The distance between the two bodies when Body 1 has fallen for \( t \) seconds is:
\[ d = s_1 - s_2 = 5t^2 - 5(t - 2)^2 \]

### Step 6: Simplifying the distance equation
Expanding \( s_2 \):
\[ s_2 = 5(t^2 - 4t + 4) = 5t^2 - 20t + 20 \]
Thus, the distance \( d \) becomes:
\[ d = 5t^2 - (5t^2 - 20t + 20) = 20t - 20 \]

### Step 7: Setting up the equation for 40 meters apart
We set the distance \( d \) equal to 40 meters:
\[ 20t - 20 = 40 \]

### Step 8: Solving for \( t \)
Adding 20 to both sides:
\[ 20t = 60 \]
Dividing by 20:
\[ t = 3 \, \text{seconds} \]

### Conclusion
The two bodies will be 40 meters apart 3 seconds after Body 1 begins to fall.