The linear momentum of a particle varies with time t as `p= a +bt + ct^(2)`. Then, whichh of the following is correct?

Options

A. Velocity of particle is inversely proportional to time
B. Displacement of the particle is independent of time
C. Forces varies with time in a quadratic manner
D. Force is independent linearly on time

Explanation

To solve the problem, we need to analyze the given expression for linear momentum \( p \) and derive the force from it. Here’s a step-by-step solution:

### Step 1: Write down the expression for linear momentum
The linear momentum of the particle is given by:
\[ p = a + bt + ct^2 \]
where \( a \), \( b \), and \( c \) are constants.

### Step 2: Differentiate the momentum with respect to time
To find the force acting on the particle, we need to calculate the rate of change of momentum with respect to time, which is given by:
\[ \frac{dp}{dt} = \frac{d}{dt}(a + bt + ct^2) \]
Since \( a \) is a constant, its derivative is zero. The derivatives of the other terms are:
\[ \frac{dp}{dt} = 0 + b + 2ct = b + 2ct \]

### Step 3: Relate the rate of change of momentum to force
According to Newton's second law, the force \( F \) acting on an object is equal to the rate of change of momentum:
\[ F = \frac{dp}{dt} \]
Thus, we can write:
\[ F = b + 2ct \]

### Step 4: Analyze the expression for force
The expression \( F = b + 2ct \) indicates that the force is a function of time \( t \). The term \( 2ct \) shows that the force changes linearly with time, assuming \( c \) is a constant.

### Conclusion
From the analysis, we conclude that the force acting on the particle is linearly dependent on time. Therefore, the correct option is:
**Option D: Force is linearly dependent on time.**

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