Formula for true force is

Options

A. F= ma
B. `F = (mdv)/(dt)`
C. `F = (d( mv))/(dt)`
D. `F = (md^(2)x)/(dt)`

Explanation

To derive the formula for true force, we start with the definition provided by Newton's second law of motion. The true force is defined as the rate of change of linear momentum. Let's go through the steps to derive the formula for true force.

### Step-by-Step Solution:

1. **Understanding Linear Momentum**:
- Linear momentum (P) is defined as the product of mass (m) and velocity (v) of an object.
- Formula: \( P = m \cdot v \)

2. **Rate of Change of Linear Momentum**:
- According to Newton's second law, the force (F) acting on an object is equal to the rate of change of its linear momentum.
- Formula: \( F = \frac{dP}{dt} \)

3. **Substituting Linear Momentum**:
- Substitute the expression for linear momentum into the equation for force.
- \( F = \frac{d(m \cdot v)}{dt} \)

4. **Applying the Product Rule**:
- Since mass (m) is constant for a given object, we can take it out of the derivative.
- Thus, we can apply the product rule of differentiation:
- \( F = m \cdot \frac{dv}{dt} + v \cdot \frac{dm}{dt} \)
- However, since \( \frac{dm}{dt} = 0 \) for constant mass, we simplify to:
- \( F = m \cdot \frac{dv}{dt} \)

5. **Recognizing Acceleration**:
- The term \( \frac{dv}{dt} \) is the definition of acceleration (a).
- Therefore, we can rewrite the equation as:
- \( F = m \cdot a \)

6. **Conclusion**:
- The true force is defined as the rate of change of linear momentum, which can be expressed as:
- \( F = \frac{dP}{dt} \)
- This is the original definition of force according to Newton's second law.

### Final Formula for True Force:
- The formula for true force is:
\[ F = \frac{dP}{dt} \]
where \( P = m \cdot v \).