The correct differential 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 find the correct differential form of Newton's second law, we can follow these steps:

### Step 1: Understand the Concept of Newton's Second Law
Newton's second law states that the force acting on an object is equal to the rate of change of its momentum. Mathematically, this can be expressed as:
\[ F = \frac{dP}{dt} \]
where \( F \) is the force, \( P \) is the momentum, and \( t \) is time.

### Step 2: Define Momentum
Momentum \( P \) is defined as the product of mass \( m \) and velocity \( v \):
\[ P = mv \]

### Step 3: Differentiate Momentum
To express the force in terms of momentum, we need to differentiate momentum with respect to time:
\[ \frac{dP}{dt} = \frac{d(mv)}{dt} \]

### Step 4: Apply the Product Rule
Using the product rule of differentiation, we can expand the derivative:
\[ \frac{d(mv)}{dt} = m \frac{dv}{dt} + v \frac{dm}{dt} \]
However, if we assume the mass \( m \) is constant (which is often the case in classical mechanics), the equation simplifies to:
\[ \frac{d(mv)}{dt} = m \frac{dv}{dt} \]

### Step 5: Relate to Force
Now, substituting back into the expression for force, we have:
\[ F = \frac{dP}{dt} = m \frac{dv}{dt} \]
Since \( \frac{dv}{dt} \) is the acceleration \( a \), we can write:
\[ F = ma \]

### Step 6: Write in Differential Form
In the differential form, we can express the force as:
\[ F = \frac{d}{dt}(mv) \]
This is the differential form of Newton's second law.

### Conclusion
From the options provided, the correct differential form of Newton's second law is:
\[ F = \frac{d}{dt}(mv) \]