Inclined planes are conveniet tools to study motion under a constant acceleration.?

Question

The time of passage of a ball rolling on an inclined plane is measured by 3 light gates positioned 60 cm apart. The ball passes the light gates at .30, 1.15, 1.70 seconds. what is the acceleration of the ball?
I am not sure how to approach this probelm. Please help explain the steps to get to the answer.

Answer 1

It travels 0.6 m in dt1 = 1.15-0.30 = 0.85s, so v1(ave) = 0.6/0.85 = 0.7059 m/s;
it is reached at t1(mid) = (0.30+1.15)/2 = 0.725 s
It travels 0.6 m in dt2 = 1.70-1.15 = 0.55s, so v2(ave) = 0.6/0.55 = 1.0909 m/s
it is reached at t2(mid) = (1.15+1.70)/2 = 1.425 s
It has a velocity change of v2(ave)-v1(ave) m/s in t2(mid)-t1(mid) s so a = (1.0909-0.7059)/(1.425-0.725) = 0.55 m/s^2.

Incidentally, one can also compute the bias in the time measurement (Δt) and the distance of the first gate (Δx), from the point when and where and when the ball was released with zero velocity. From these the other velocities are easily obtained.
v1(start) = v1(ave) - a*(t2-t1)/2 = 0.472 m/s
Δt = t1 - v1(start)/a = -0.558 s
Δx = a(t1-Δt)^2/2 = 0.203 m

Answer 2

hmmm, i think you have to average the accelerations

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