Here's one way of doing it. You need 3 gates.
You need the distance between the gates, say S
1 between the 1st two and S
2 between the 2nd and 3rd.
Time between 1st two is t
1 and between 2nd and 3rd is t
2Those are your measurements.
Here is a graph of what's happening. It's a velocity-time graph showing the times and distances travelled (area under). It also shows the velocities at the start and end of each section. V
1 V
2 and V
3You know that the distance travelled in a section is average velocity times time, and that average velocities are (V1 + V2) / 2 in S1 and (V2 + V3) / 2 in S2
The times are t2 - t1 and t3 - t2
Average velocity = distance / time
So
average velocity in first section = S1 / (t2 - t1)Average velocity in 2nd section = S2 / (t3 - t2)These are all things you have measured in the experiment.
The average velocity in the 1st section is marked as U on the graph. It is the velocity at exactly half the time interval between t1 and t2. (For uniform acceleration)
The average velocity in the 2nd section is marked as V on the graph and occurs at exactly half the time interval there.
These are marked with the dotted lines.
So to find the acceleration you need to apply V = U + gT
g = (V - U) / T [acceleration is change in velocity divided by time taken]
T is the time interval between those two halfway points. It's marked on the graph. Can you work out what it is?
If so, you can now find the acceleration because you can calculate V and U from your measurements as explained above in bold type.