Answer
We can rank the fragments in order of the magnitude of their momentum, from smallest to largest:
$d = e \lt a = b \lt c$
Work Step by Step
(a) We can find the speed:
$\frac{1}{2}mv^2 = 400~J$
$v = \sqrt{\frac{(2)(400~J)}{m}}$
$v = \sqrt{\frac{(2)(400~J)}{8~kg}}$
$v = 10~m/s$
We can find the momentum:
$p = mv = (8~kg)(10~m/s) = 80~kg~m/s$
(b) We can find the speed:
$\frac{1}{2}mv^2 = 1600~J$
$v = \sqrt{\frac{(2)(1600~J)}{m}}$
$v = \sqrt{\frac{(2)(1600~J)}{2~kg}}$
$v = 40~m/s$
We can find the momentum:
$p = mv = (2~kg)(40~m/s) = 80~kg~m/s$
(c) We can find the speed:
$\frac{1}{2}mv^2 = 1600~J$
$v = \sqrt{\frac{(2)(1600~J)}{m}}$
$v = \sqrt{\frac{(2)(1600~J)}{4~kg}}$
$v = 28.28~m/s$
We can find the momentum:
$p = mv = (4~kg)(28.28~m/s) = 113~kg~m/s$
(d) We can find the speed:
$\frac{1}{2}mv^2 = 100~J$
$v = \sqrt{\frac{(2)(100~J)}{m}}$
$v = \sqrt{\frac{(2)(100~J)}{16~kg}}$
$v = 3.54~m/s$
We can find the momentum:
$p = mv = (16~kg)(3.54~m/s) = 57~kg~m/s$
(e) We can find the speed:
$\frac{1}{2}mv^2 = 1600~J$
$v = \sqrt{\frac{(2)(1600~J)}{m}}$
$v = \sqrt{\frac{(2)(1600~J)}{1~kg}}$
$v = 56.57~m/s$
We can find the momentum:
$p = mv = (1~kg)(56.57~m/s) = 57~kg~m/s$
We can rank the fragments in order of the magnitude of their momentum, from smallest to largest:
$d = e \lt a = b \lt c$