12. A particle of mass $$ m $$ falls against a resistance, per unit mass, proportional to its velocity $$ v $$. Given that $$ k0 $$, the terminal velocity is A $$ k v $$ B $$ \frac{1}{s} $$ C $$ \frac{ \pi}{k} $$ D kg

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12. A particle of mass $$ m $$ falls against a resistance, per unit mass, proportional to its velocity $$ v $$. Given that $$ k>0 $$, the terminal velocity is A $$ k v $$ B $$ \frac{1}{s} $$ C $$ \frac{ \pi}{k} $$ D kg

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To find the terminal velocity of a particle falling against a resistance proportional to its velocity, we can start by analyzing the forces acting on the particle.

1. **Identify the forces:**
   - The weight of the particle acting downward: $$ F_g = mg $$
   - The resistance force acting upward, which is proportional to the velocity: $$ F_r = kv $$

2. **Set up the equation of motion:**
   At terminal velocity, the net force acting on the particle is zero. Therefore, we can set the weight equal to the resistance force:
   $$
   mg = kv_t
   $$
   where $$ v_t $$ is the terminal velocity.

3. **Solve for terminal velocity:**
   Rearranging the equation gives:
   $$
   v_t = \frac{mg}{k}
   $$

4. **Analyze the options:**
   - A $$ kv $$: This does not represent terminal velocity.
   - B $$ \frac{1}{s} $$: This does not represent terminal velocity.
   - C $$ \frac{\pi}{k} $$: This does not represent terminal velocity.
   - D kg: This does not represent terminal velocity.

None of the options provided directly match the derived expression for terminal velocity $$ \frac{mg}{k} $$.

However, if we consider the context of the problem, the terminal velocity is typically expressed in terms of the parameters involved. Since $$ m $$ and $$ g $$ are constants, the terminal velocity is dependent on the mass and the resistance coefficient $$ k $$.

Thus, the correct expression for terminal velocity is $$ \frac{mg}{k} $$, which is not listed among the options. Therefore, it seems there may be an error in the options provided.

If you have any additional context or constraints, please provide them for further clarification.
The terminal velocity is $$ \frac{mg}{k} $$, which is not among the provided options.

12. A particle of mass \( m \) falls against a resistance, per unit mass, proportional to its velocity \( v \). Given that \( k>0 \), the terminal velocity is A \( k v \) B \( \frac{1}{s} \) C \( \frac{ \pi}{k} \) D kg

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