The correct from of Newton`s second law is

Options

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

Answer

To solve the question regarding the correct form of Newton's second law, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Newton's Second Law**: Newton's second law states that the rate of change of momentum of an object is directly proportional to the net force acting on it. This can be expressed mathematically as:
\[ F \propto \frac{dp}{dt} \]
where \( F \) is the force, \( p \) is the momentum, and \( t \) is time.

2. **Define Momentum**: Momentum \( p \) is defined as the product of mass \( m \) and velocity \( v \):
\[ p = mv \]

3. **Calculate the Rate of Change of Momentum**: To find the rate of change of momentum, we differentiate momentum with respect to time:
\[ \frac{dp}{dt} = \frac{d(mv)}{dt} \]

4. **Apply the Product Rule**: If mass \( m \) is constant, we can apply the product rule of differentiation:
\[ \frac{dp}{dt} = m \frac{dv}{dt} + v \frac{dm}{dt} \]
Since \( m \) is constant, \( \frac{dm}{dt} = 0 \), thus:
\[ \frac{dp}{dt} = m \frac{dv}{dt} \]

5. **Relate to Force**: According to Newton's second law, we can equate the force \( F \) to the rate of change of momentum:
\[ F = \frac{dp}{dt} \]
Therefore, substituting the expression we derived:
\[ F = m \frac{dv}{dt} \]

6. **Recognize Acceleration**: The term \( \frac{dv}{dt} \) is defined as acceleration \( a \):
\[ a = \frac{dv}{dt} \]
Thus, we can rewrite the equation as:
\[ F = ma \]

7. **General Form**: The general form of Newton's second law, which applies even when mass is not constant, is:
\[ F = \frac{dp}{dt} \]
This is the most general form of Newton's second law.

8. **Identify the Correct Option**: Based on the options provided, the correct form of Newton's second law is the one that states:
\[ F = \frac{dp}{dt} \]
or in the special case of constant mass:
\[ F = ma \]

### Conclusion:
The correct form of Newton's second law is \( F = \frac{dp}{dt} \) or \( F = ma \) when mass is constant.

---