A particle of mass \(m\) is initially at rest on a horizontal, frictionless surface at time \(t = 0\). Starting at \(t = 0\), a horizontal force \(F(t) = F_0 e^{-t/\tau}\) is applied to the particle, where \(F_0\) and \(\tau\) are positive constants.

Choice	Final Momentum \(p_{\infty}\)	Fraction of Total Impulse (\(0 \le t \le \tau\))
A	\(\dfrac{F_0}{\tau}\)	\(e^{-1}\)
B	\(\dfrac{F_0}{\tau}\)	\(1 - e^{-1}\)
C	\(F_0 \tau\)	\(e^{-1}\)
D	\(F_0 \tau\)	\(1 - e^{-1}\)

Which choice correctly identifies the magnitude of the final momentum \(p_{\infty}\) of the particle as \(t \to \infty\) and the fraction of the total impulse delivered to the particle during the time interval \(0 \le t \le \tau\)?