To be quantitative I launch slides with different mass using a spring
mechanism on this air-track. The launch mechanism ensures that
I can reproduce the magnitude of the force and distance over which it acts
for slides with different mass. Rather than measure the acceleration,
I measure the velocity v at a specific distance
from the beginning of the motion with a photo-gate. From our study
of kinematics recall that v is related to the acceleration and
distance covered by the following equation
Since I shall start from rest I have 
and hence the formula simplifies
to
Now according to Newton's second law
we have
Inserting this in the previous equation yields
Now it happens that the way our photo-gate works is by measuring
the time taken for the cart to pass the gate which is related to
the velocity as
where 
is the length of the slide. Inserting the expression for
v here yields the following relation between the
time registered on the photo-gate and the mass of the
slide:
I don't know details about the force and distance over which it
acts in the launching mechanism, just that I can reproduce
F and d between launches. All I want to do is to check that
grows in proportion to 
. The unloaded cart
has a mass of 
kg while the loaded cart has a mass
kg. The ratio of times should therefore be
I measure
and hence we predict
for the loaded cart. We do the experiment on the loaded cart and
get a reading close to this.