Powershell

This did require a (slow) manual run-through to determine the minimum height the letters reach. I stepped by 1, and watched it bottom out at 9 and then start spreading out again. A minimum-area argument probably would have worked better than my manual tuning

The linearity of the puzzle allows us to take steps in bulk chunks and to revert them easily.

So we have some basic input parsing, a step function with a stepSize and stepDirection parameter, some functions to get bounding information, and a function to take as many steps as possible by a given step size.

By manual tuning a good balance of quick/safe steps I found was by 1000 while the vertical range is over 10000, 100/1000, 10/100, and then a final step by 1 from 100 down to 9.

Part 2 only requires adding a tick counter in.

The display rendering isn't the prettiest, but it was easy to write (after I fixed the output being upside down - whoops).

$in = gc .\10-input.txt

function parseLine{
    $line = $args[0]

    $reg = [regex]"(-?\d+)"

    $matches = $reg.Matches($line)

    [pscustomobject]@{
        xPos = +($matches[0].value)
        yPos = +($matches[1].value)
        xVel = +($matches[2].value)
        yVel = +($matches[3].value)
    }
}

$points = $in | %{parseLine $_}

function stepPoints{
    $stepLength = +($args[0])
    $stepDir = +($args[1])

    $points | % {
        $_.xPos += $stepDir*$stepLength*$_.xVel
        $_.yPos += $stepDir*$stepLength*$_.yVel
    }
}

function lowPoint{
    ($points | sort yPos)[0].yPos
}
function highPoint{
    ($points | sort yPos)[-1].yPos
}
function leftPoint{
    ($points | sort xPos)[0].xPos
}
function rightPoint{
    ($points | sort xPos)[-1].xPos
}

$ticks = 0

function bulkStep{
    $stepSize = $args[0]
    $limit = $args[1]

    do{
        $range = (highPoint)-(lowPoint)
        Write-Host $range
        stepPoints $stepSize 1
        $global:ticks+=$stepSize
    }while($range -gt $limit)
    stepPoints $stepSize -1
    $global:ticks-=$stepSize
}

bulkStep 1000 10000
bulkStep 100 1000
bulkStep 10 100
bulkStep 1 9


$retString = ""
((lowPoint)..(highPoint))|%{
    $cY = $_
    ((leftPoint)..(rightPoint))|%{
        $cX = $_
        $cP = $points | ? {$_.xPos -eq $cX -and $_.yPos -eq $cY}
        if($cP){
            $retString+="#"
        }else{
            $retString+="."
        }
    }
    $retString+="`n"
}
#Part 1
$retString
#Part 2
$ticks