1. ## constant acceleration

if a ball is placed on a straight sloping track and then released from rest, the distances that it moves in successive equal intervals of time are found to be in the ratio 1:3:5:7...
show that this is consistent with the theory that the ball rolls down the track with constant acceleration

what is the best way to do this? personnally i would try to prove that the accelerations over the four periods of time considered are equal, but is this appropriate? would it be useful to compare these to acceleration over the total distance considered 16D? or could one use analytical geometry (a=gradient)?

2. Hi

Suppose the acceleration is constant
I am using a system of coordinates (Ox) such that O is the point where the ball is released from rest, (Ox) following the slope of the track

$\displaystyle a_x = a$

$\displaystyle v_x = at$

$\displaystyle x = \frac12 at^2$

For t=0, x=0
For t=T, x=0.5atē
For t=2T, x=2atē
For t=3T, x=4.5atē

From t=0 to t=T the distance is 0.5atē
From t=T to t=2T the distance is 1.5atē = 3 x 0.5atē
From t=2T to t=3T the distance is 2.5atē = 5 x 0.5atē

The ratio is 1:3:5 etc