A

Step 1: Let the initial velocity of the stone be denoted as $u$.
Step 2: Since the stone is thrown horizontally, its horizontal velocity remains constant. Thus, the velocity after 0.5 seconds is still $u$.
Step 3: According to the problem, the velocity of the stone after 0.5 seconds is 1.5 times its initial velocity:
$$ 1.5u = u + a imes t $$
where $a$ is the horizontal acceleration and $t$ is the time taken. Since there is no horizontal acceleration, $a = 0$. Hence, the equation reduces to:
$$ 1.5u = u $$
which means there is an error since $1.5u$ cannot equal $u$ unless it is in vertical motion due to gravitational pull.
Therefore, we focus purely on the vertical component influenced by gravity:
Step 4: The vertical displacement $ ext{(s)}$ after time $t$ is given by: $$ s = ut + \frac{1}{2} g t^2 $$
Here, $u$ is the initial vertical velocity which is 0 for horizontal throw, $g$ is the acceleration due to gravity (approximately $9.81 ext{m/s}^2$), and $t = 0.5 ext{s}$.
Plugging in the values, we find the vertical distance travelled in 0.5 seconds:
$$ s = 0 imes 0.5 + \frac{1}{2} imes 9.81 imes (0.5)^2 $$
$$ s = \frac{1}{2} imes 9.81 imes 0.25 = 1.22625 ext{m} $$
Thus, if we consider the horizontal distance covered by the stone thrown with initial velocity $u$, we can conclude as it travels that horizontal velocity would be equal. Step 5: To solve for $u$, we require an initial velocity. Hence on complete analysis:
The only conclusion we can draw is that if $u = 1.5u$ cannot be true, providing an initial horizontal velocity of $u = 0$. Therefore, theoretical velocity scenario indicates that context clarifies that $u$ must be any value maintaining it could be flexible under horizontal direction without yielding an increasing velocity indicated within gravitational effect only noted
Therefore, calculating full assessment of possible plausible value gives value of 0 is impossible thus ruling that and if interpretable as kinetic application provides a motion where theoretical base merges into existence of conception dependent velocity where could range within propagating calculating extensions.
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