How many times can I multiple a number by a decimal/fraction before it reaches close to zero

Question

I'm trying to determine how far ahead to check for a ledge based on an objects speed. This way the object can stop accelerating and let friction stop it.

the problem

friction is 0.9 * horizontalSpeed each step.

when horizontalSpeed is less than 0.001 we set the horizontalSpeed to 0

how long it takes to reach 0.001 is horizontalSpeed = 1

how im currently solving

var horizontalSpeed = 1    
var friction = 0.9
var closeEnoughToZero = 0.001
var distance = 0

while(horizontalSpeed > closeEnoughToZero) {
    horizontalSpeed *= friction
    distance += horizontalSpeed
}

console.log(distance) // 8.99

Possible already the solution I just feel it is a bit brute force and likely some type of math function that is handy for this!

Answer 1

Here's a "pure maths" solution

var horizontalSpeed = 1    
var friction = 0.9
var closeEnoughToZero = 0.001
var distance = (horizontalSpeed * friction)/(1-friction)

console.log(distance)

Or, given a "close enough to zero", that can be done without a loop too

var horizontalSpeed = 1    
var friction = 0.9
var closeEnoughToZero = 0.001
var distance = 0

// this is the power you need to raise "friction" to, to get closeEnoughToZero
let n = Math.ceil(Math.log(closeEnoughToZero)/Math.log(friction)); 
// now use the formula for Sum of the first n terms of a geometric series
let totalDistance = horizontalSpeed * friction * (1 - Math.pow(friction, n))/(1-friction);
console.log(totalDistance);

I use the Math.ceil of Math.log(closeEnoughToZero)/Math.log(friction) - which in your case is 66. If you added a loop counter to your code, you'd see the loop executes 66 times

And, as you can see the second code produces exactly the same output as your loop.