Answer
The vertical component of displacement is 0.15 meters.
The horizontal component of displacement is 0.90 meters.
Work Step by Step
Let $m$ be the mass of a projectile and let $M$ be the mass of the pendulum. Let $v$ be the initial speed of a projectile.
We can use conservation of momentum to find the speed $v'$ just after the collision:
$m~v = (m+M)~v'$
We can use conservation of energy to find the height $h$ reached by the pendulum as it swings up:
$\frac{1}{2}(m+M)~(v')^2 = (m+M)~gh$
$v' = \sqrt{2gh}$
We can replace $v'$ in the first equation above:
$m~v = (m+M)\sqrt{2gh}$
$h = (\frac{m}{m+M})^2 \frac{v^2}{2g}$
$h = (\frac{0.028~kg}{0.028~kg+3.1~kg})^2 \frac{(190~m/s)^2}{(2)(9.80~m/s^2)}$
$h = 0.15~m$
Using the Pythagorean theorem, we can find the horizontal displacement $x$.
$x^2 + (L-h)^2 = L^2$
$x = \sqrt{2Lh - h^2}$
$x = \sqrt{(2)(2.8~m)(0.15~m) - (0.15~m)^2}$
$x = 0.90~m$
The vertical component of displacement is 0.15 meters.
The horizontal component of displacement is 0.90 meters.
