Answer
(a) In each of the three cases, the final speed is $\sqrt{v_0^2+2gh}$. Thus all three situations result in the same final speed.
(b) The final speed for each rock is $19.8~m/s$
Work Step by Step
(a) Let down be the positive vertical direction. Let $h$ be the height of the cliff. Let $v_0$ be the rock's initial velocity. Note that $a_y = g$.
We can find the final vertical velocity of the rock that is thrown straight down:
$v_{y,f}^2 = v_0^2+2gh$
$v_{y,f} = \sqrt{v_0^2+2gh}$
Since the horizontal component of velocity is zero, the speed just before hitting the ground is $\sqrt{v_0^2+2gh}$
We can find the final vertical velocity of the rock that is thrown straight up:
$v_{y,f}^2 = (-v_0)^2+2gh$
$v_{y,f} = \sqrt{(-v_0)^2+2gh}$
$v_{y,f} = \sqrt{v_0^2+2gh}$
Since the horizontal component of velocity is zero, the speed just before hitting the ground is $\sqrt{v_0^2+2gh}$
We can find the final vertical velocity of the rock that is thrown horizontally:
$v_{y,f}^2 = v_0^2+2gh$
$v_{y,f} = \sqrt{0+2gh}$
$v_{y,f} = \sqrt{2gh}$
We can find the final speed of the rock:
$v_f = \sqrt{v_0^2+v_{y,f}^2}$
$v_f = \sqrt{v_0^2+(\sqrt{2gh})^2}$
$v_f = \sqrt{v_0^2+2gh}$
In each of the three cases, the final speed is $\sqrt{v_0^2+2gh}$. Thus all three situations result in the same final speed.
(b) We can find the final speed of each of the three rocks:
$v_f = \sqrt{v_0^2+2gh}$
$v_f = \sqrt{(10.0~m/s)^2+(2)(9.80~m/s^2)(15.0~m)}$
$v_f = 19.8~m/s$
The final speed for each rock is $19.8~m/s$
