A particle of mass m is moving in a straight line with momentum p. Starting at time $$t = 0$$, a force $$F = kt$$ acts in the same direction on the moving particle during time interval T so that its momentum changes from p to 3p. Here k is a constant. The value of T is

We begin by recalling the relation between force and the change in momentum. The total impulse imparted to a particle in a given time interval equals the integral of the force with respect to time over that interval. Stated mathematically, the impulse $$J$$ over the interval from $$t = 0$$ to $$t = T$$ is given by the formula

$$J \;=\; \int_{0}^{T} F \, dt.$$

The problem tells us that the force varies with time according to $$F = k t,$$ where $$k$$ is a constant and $$t$$ is the time measured from zero. Substituting this expression for $$F$$ in the impulse formula, we have

$$J \;=\; \int_{0}^{T} k t \, dt.$$

Because $$k$$ is a constant, it can be taken outside the integral sign:

$$J \;=\; k \int_{0}^{T} t \, dt.$$

Next, we evaluate the elementary integral of $$t$$ with respect to $$t$$. We use the standard result

$$\int t \, dt \;=\; \frac{t^{2}}{2} + C,$$

where $$C$$ is the constant of integration. Applying the definite-integral limits from $$0$$ to $$T,$$ the constant $$C$$ cancels out, giving

$$\int_{0}^{T} t \, dt \;=\; \left[\frac{t^{2}}{2}\right]_{0}^{T} \;=\; \frac{T^{2}}{2} - \frac{0^{2}}{2} \;=\; \frac{T^{2}}{2}.$$

Substituting this result back into the impulse expression, we obtain

$$J \;=\; k \left(\frac{T^{2}}{2}\right) \;=\; \frac{k T^{2}}{2}.$$

Now we turn to the definition of impulse in terms of momentum. The impulse delivered to the particle equals the change in its momentum. Initially the momentum is given as $$p,$$ and after the force has acted for the time interval $$T,$$ the momentum becomes $$3p.$$ Therefore, the change in momentum, denoted $$\Delta p,$$ is

$$\Delta p \;=\; 3p - p \;=\; 2p.$$

Setting the impulse equal to this change in momentum gives

$$\frac{k T^{2}}{2} \;=\; 2p.$$

We wish to solve this equation for the time $$T.$$ First, multiply both sides by $$2$$ to eliminate the denominator:

$$k T^{2} \;=\; 4p.$$

Next, divide both sides by $$k$$ to isolate $$T^{2}$$:

$$T^{2} \;=\; \frac{4p}{k}.$$

Finally, take the positive square root of both sides (time is positive) to obtain $$T$$:

$$T \;=\; 2 \sqrt{\frac{p}{k}}.$$

Comparing this expression with the options supplied, we see that it exactly matches Option B:

$$2\sqrt{\frac{p}{k}}.$$

Hence, the correct answer is Option B.