The linear momentum P of a body moving in one dimension varies with time according to the equation `P = at^(3) + "bt"` where a and b are positive constants. The net force acting on the body is :

Options

A. Proportional to `t^(2)`
B. A constant
C. Proportional to t
D. Inversely proportional to it

Answer

To solve the problem, we need to find the net force acting on a body whose linear momentum \( P \) varies with time according to the equation:

\[ P = at^3 + bt \]

where \( a \) and \( b \) are positive constants.

### Step-by-step Solution:

1. **Understand the relationship between force and momentum**:
The net force \( F \) acting on a body is defined as the rate of change of momentum with respect to time. Mathematically, this is expressed as:
\[ F = \frac{dP}{dt} \]

2. **Differentiate the momentum equation**:
We need to differentiate the given momentum equation \( P = at^3 + bt \) with respect to time \( t \):
\[ \frac{dP}{dt} = \frac{d}{dt}(at^3 + bt) \]

3. **Apply the differentiation**:
Using the power rule of differentiation:
- The derivative of \( at^3 \) is \( 3at^2 \).
- The derivative of \( bt \) is \( b \).
Therefore, we have:
\[ \frac{dP}{dt} = 3at^2 + b \]

4. **Express the net force**:
Substitute the result from the differentiation into the equation for force:
\[ F = 3at^2 + b \]

5. **Conclusion**:
The net force acting on the body is given by:
\[ F = 3at^2 + b \]

### Final Answer:
The net force acting on the body is \( F = 3at^2 + b \).

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