通过距离和摩擦计算速度

Question

I am writing a game in Javascript, Canvas, HTML5, and I just found a big problem, related to advanced maths. The game is in flat 2D, so you see the world from over. That means there is no gravity, only friction.

CODE:

var friction = 0.97

var target = {
  x: 70,
  y: 90,
}

var ball = {
  x: 10,
  y: 20,
  vx: 0,
  vy: 0,
}
var dx = ball.x - target.x
var dy = ball.y - target.y
var angle = Math.atan2(dy, dx)
var dist = Math.sqrt(dx * dx + dy * dy) // <--- AS FAR AS I'VE COME
var speed = ??? // <--- HERE IS THE PROBLEM
ball.vx = Math.cos(angle) * speed
ball.vy = Math.sin(angle) * speed

function update() {
  ball.vx *= friction
  ball.vy *= friction
  ball.x += ball.vx
  ball.y += ball.vy

  // Drawing and other unneccesary things to involve here

  window.requestAnimationFrame(update)
}

update()

So, my question is: How do I calculate the speed in the section where I create the ball, to make the ball stop at exactly the targets x and y position? As far as I've come I've only calculated the starting distance between the ball and the targets position.. Soo, any ideas? Is there an equation for this?

Answer 1

So you want the sum for i from 1 to ∞ over the term (speed × friction ⁱ ), and you want the result to equal dist. This is an infinite geometric series , the limit of which is speed×friction/(1 − friction).

So simply solve the equation for speed, and you obtain:

speed = (1/friction - 1)*dist