To solve the problem, we need to calculate the work done against air resistance when a bullet is fired horizontally from a height. Here’s the step-by-step solution:
### Step 1: Identify the given data
- Mass of the bullet, \( m = 10 \, \text{g} = 0.01 \, \text{kg} \) (convert grams to kilograms)
- Initial velocity of the bullet, \( u = 1000 \, \text{m/s} \)
- Final velocity of the bullet, \( v = 500 \, \text{m/s} \)
- Height from which the bullet is fired, \( h = 50 \, \text{m} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)
### Step 2: Calculate the initial energy (E_initial)
The initial energy consists of kinetic energy (due to its horizontal motion) and potential energy (due to its height).
\[
E_{\text{initial}} = \text{K.E.} + \text{P.E.} = \frac{1}{2} m u^2 + mgh
\]
Substituting the values:
\[
E_{\text{initial}} = \frac{1}{2} \times 0.01 \, \text{kg} \times (1000 \, \text{m/s})^2 + 0.01 \, \text{kg} \times 10 \, \text{m/s}^2 \times 50 \, \text{m}
\]
Calculating each term:
1. Kinetic Energy:
\[
\frac{1}{2} \times 0.01 \times 1000000 = 5000 \, \text{J}
\]
2. Potential Energy:
\[
0.01 \times 10 \times 50 = 5 \, \text{J}
\]
Thus,
\[
E_{\text{initial}} = 5000 \, \text{J} + 5 \, \text{J} = 5005 \, \text{J}
\]
### Step 3: Calculate the final energy (E_final)
The final energy is only kinetic energy since the bullet has reached the ground (potential energy is zero at ground level).
\[
E_{\text{final}} = \frac{1}{2} m v^2
\]
Substituting the values:
\[
E_{\text{final}} = \frac{1}{2} \times 0.01 \, \text{kg} \times (500 \, \text{m/s})^2
\]
Calculating:
\[
E_{\text{final}} = \frac{1}{2} \times 0.01 \times 250000 = 1250 \, \text{J}
\]
### Step 4: Calculate the work done against air resistance
The work done against air resistance is the difference between the initial energy and the final energy.
\[
W = E_{\text{initial}} - E_{\text{final}}
\]
Substituting the values:
\[
W = 5005 \, \text{J} - 1250 \, \text{J} = 3755 \, \text{J}
\]
### Final Answer
The work done against air resistance in the trajectory of the bullet is \( \boxed{3755 \, \text{J}} \).
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