Galileo used marbles rolling down inclined planes to deduce some basic properties of constant accelerated motion. In particular, he measured the distance a marble rolled during specific time periods. For example, suppose a marble starts from rest and begins rolling down an inclined plane with constant acceleration a. After 1 s, you find that it moved a distance . a. In terms of x, how far does it move in the next 1 s time period—that is, in the time between 1 s and 2 s?
b. How far does it move in the next second of the motion?
c. How far does it move in the nth second of the motion?

a)  y₁ = ½ a, b) y₂ = 4 y₁, c)  y₃ = 9 y₁

Explanation:

For this exercise we can use the accelerated motion relationships.

Let's set a reference system where the x axis is parallel to the plane and its positive side is going down the plane.

          y = y₀ + v₀ t + ½ a t²

in that case where we throw the marble is the zero point, y₀ = 0, as part of rest its initial velocity is zero v₀ = 0 and a  is the acceleration along the inclined plane

           y = ½ a t²

a) in the first second t = 1

            y₁ = ½ a

b) in the next second of movement

           t = 2 s

           y₂ = ½ a 2²

           y₂ = 4 ½ a

           y₂ = 4 y₁

c) for the next second

            t = 3 s

            y₃ = ½ a 3²

            y₃ = 9 ½ a

            y₃ = 9 y₁

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