28th August 2005 - 09:10 PM
To make it easier to see, I first graphed it in excel.
1. The trolly would stop accelerating if the velocity would stop changing, or on the graph, the line would become straight at a 45 degree angle to the base. I can't really tell without more measurements, but it looks like towards the end, it does seem like it stopped accelerating.
2. First of all you need to find the change in velocity before you can find the acceleration. I assume you don't have to come up with an equation to fit your data points, so I don't think you'll need the differential equations, but I'll throw them in anyways. They are useful if you need to find the exact velocity or acceleration at a certain time, not just an average like what I'll be doing below.
Velocity = change in distance / change in time
v=d/dt(distance) where t=time in sec (d/dt means take the derivative of equation with respect to variable 't' which is time in our case)
Acceleration = change in velocity / change in time
To find the velocity of each time interval we'll use this equation instead of using derivatives because your points are taken at each discreet time intervals. Keep in mind that this is an average velocity not the exact velocity, but since the time interval is 0.1 sec, it should be a good approximation. It doesn't mean that right at zero second, the trolly is going 53cm/sec because right at 0sec, it is going 0cm/sec.
V=(distance final - distance initial) / (time final - time initial)
Thus for the first sample:
V0=( 5.3cm - 0cm ) / ( 0.1sec - 0sec)
V0= 53 cm/sec
For the second sample:
V1=( 11.5cm - 5.3cm ) / (0.2sec - 0.1sec)
and so fourth for V2, V3, and V4
It works out that:
V0= 53 cm/sec
V1= 62 cm/sec
V2= 101 cm/sec
V3= 125 cm/sec
V4= 151 cm/sec
(verify for yourself)
Now that you have the velocities at each time interval, you can find the acceleration:
Thus A= (velocity final - velcoity initial) / (time final - time initial)
A0=(53 cm/sec - 0cm/sec) / (0.1sec - 0sec)
The rest I'll leave up to you to do.
3. To find the average, it is easy, just add up all the acceleration points then divide by the number of points you added up.
Ave = (K1+K2+...+Kn)/n where n is an integer
Ave = (2+3+1)/3 = 2
4. The graph is telling you the distance traveled compared to the amount of time it took to travel that distance. From this graph acceleration occurs when the distance vs time rises or decreases . If the graph starts to become linear ( the line makes a 45 degree angle to the base of the graph ) then it has stopped accelerating and has constant speed. This is what seems to be happening towards the end of your measurements as friction and drag starts to counter the acceleration due to gravity.
Hope this helps a little. What's this project for?