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The linear momentum P of a particle vari...

The linear momentum P of a particle varies with time as follows `P=a+bt^(2)`
Where a and b are constants. The net force acting on the particle is

A

proportional to t

B

proportional to `t^(2)`

C

Zero

D

constant

Text Solution

AI Generated Solution

The correct Answer is:
To find the net force acting on a particle whose linear momentum \( P \) varies with time as \( P = a + bt^2 \), we can follow these steps: ### Step 1: Understand the relationship between force and momentum The net force \( F \) acting on a particle is given by the rate of change of momentum with respect to time: \[ F = \frac{dP}{dt} \] ### Step 2: Differentiate the momentum function Given the momentum function: \[ P = a + bt^2 \] we need to differentiate this with respect to time \( t \). ### Step 3: Apply differentiation Differentiating \( P \) with respect to \( t \): - The derivative of a constant \( a \) is \( 0 \). - The derivative of \( bt^2 \) with respect to \( t \) is \( 2bt \). Thus, we have: \[ \frac{dP}{dt} = 0 + 2bt = 2bt \] ### Step 4: Write the expression for net force Now substituting back into the equation for force: \[ F = \frac{dP}{dt} = 2bt \] ### Conclusion The net force acting on the particle is: \[ F = 2bt \]
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