We are learning the conservation of energy explorations. (roller coaster) Marble

Question

We are learning the conservation of energy explorations. (roller coaster) Marble diameter = .019m

Locations: Height(m), Photogate time(s)
30: 1.24, .0123

60: .25, .0130

90: .313, .0192

120: .12, .0094

I need help on finding the theoretical and actual velocities, for then which I have to find the % error of each location, but that is simple until I get the other two factors. I used v = 2gh for theoretical velocity, but I'm not sure if that's correct or not! Please help, I need help with theoretical and actual velocities of the above information!

Explanation / Answer

.5mv^2 = mgh
so v = sqrt(2gh).
yes you're correct. g = 9.8m/s

Here's my lab from a year ago, i still have it: (we used a toy car instead of a marble) also read the conclusion part first- at the bottom . hope this helps!


Purpose:

What variables will effect the motion of the cart going down the ramp?
We think there are two variables that might effect the motion of the cart. These are mass and elevation. We will test this by determining the relationship between time and distance traveled for a control car, and then determining the same relationship for a heavier cart and for a higher elevation

Apparatus:
2 stands
2 photogates
Black box
Ramp
Computer with Data studio installed
Stickie note
Meter stick
3 physics textbooks
.5 kg mass
.5 kg frictionless cart



Procedure:
If not done beforehand the apparatus must be set up. After getting all the required materials, begin by placing two books on one end of the table.

Place the ramp on these so that the end of the ramp is just over the books and the other end is still on the table. Mark zero position as the place where the ramp meets the books.

Attach the photogates to their stands. Place one stand at zero position and the other at 10 centimeters down the ramp from zero.

Plug the photogates into the black box. Plug the black box into the computer. Finally, plug the black box into the wall. Boot up the computer and start the software. Set up the experiment in DataStudio as explained by the teacher. Attach a sticky note to the front of the cart.
Record the position of the second photogate. Position is your dependent variable.
Place the cart on the top of the ramp so that the sticky note is just behind the first photogate. Let go of cart and catch it after passing through the second photogate.
If the setup was done correctly the time should be displayed on the computer screen. Record this time in your notebook. Time is the independent variable.
Repeat 3 and 4 two more times so that you have three data points.
Move the second photogate 10 centimeters further away from the first.
Repeat steps 2-6 until you have completed a one meter trial. You should have ten trials completed.
This was the control group. Record this by labeling the data table ‘Control’. The constants in this experiment were mass (.5 kg) and elevation (two textbooks).
For the change in mass experiment, repeat 1-7 with one difference: place a .5 kg mass on the cart and run all tests this way. Label this table “Change in Mass”. Elevation (two textbooks) and mass (1 kg) were constant in this experiment.
For the change in height experiment, repeat 1-7 with one difference: place a third book in between the others and the ramp. Do not use the mass in this experiment; if it is still in the cart, remove it. Label this table “Change in Height”. The constants in this experiment were mass (.5 kg) and elevation (three textbooks).


Data:

Explanation of variables in the data tables:

Time is the amount of time elapsed in seconds.
Position is the distance from zero in meters.

For more accurate results, an average is taken of the three trials for each part of the experiment.
Example: Average position=5.686m+5.609+5.538=16.833/3= 5.611m. The average(if necessary) is rounded to the nearest thousandth of a second.

Control
.5 kg cart, height of two textbooks (0.085 m)
Position (m) Time (s) Trial 1 Time (s) Trial 2 Time (s) Trial 3
0.10 0.6061 0.6238 0.6006
0.20 0.8086 0.8169 0.8116
0.30 1.0248 1.0189 1.0265
0.40 1.2238 1.2264 1.2228
0.50 1.3883 1.3744 1.3648
0.60 1.4736 1.4797 1.4968
0.70 1.6856 1.6763 1.7043
0.80 1.7826 1.7751 1.8023
0.90 1.9212 1.9179 1.8983
1.0 2.0348 2.0238 2.0323

Position (m) Average time (s)
0.100 0.6102
0.200 0.8124
0.300 1.0234
0.400 1.2243
0.500 1.3758
0.600 1.4834
0.700 1.6887
0.800 1.7867
0.900 1.9125
1.00 2.0303

Position (m) Average time2
(s2)
0.100 .3723
0.200 .6600
0.300 1.0473
0.400 1.4989
0.500 1.8928
0.600 2.2005
0.700 2.8517
0.800 3.1923
0.900 3.6577
1.00 4.1221


Change in Mass
1 kg cart, height of two textbooks (.085 m)
Position (m) Time (s) Trial 1 Time (s) Trial 2 Time (s) Trial 3
0.100 0.5377 0.5474 0.5331
0.200 0.8200 0.8170 0.8124
0.300 1.0586 1.0304 1.0503
0.400 1.2064 1.2140 1.1929
0.500 1.3478 1.3579 1.3792
0.600 1.5161 1.4974 1.5288
0.700 1.5956 1.5754 1.5826
0.800 1.7302 1.7573 1.7505
0.900 1.8584 1.8308 1.8709
1.00 1.9993 2.0359 2.0510

Position (m) Average time (s)
0.100 0.5394
0.200 0.8165
0.300 1.0464
0.400 1.2044
0.500 1.3616
0.600 1.5141
0.700 1.5845
0.800 1.746
0.900 1.8534
1.00 2.0284

Position (m) Average time2
(s2)
0.100 .2910
0.200 .6667
0.300 1.0950
0.400 1.4506
0.500 1.8540
0.600 2.2925
0.700 2.5106
0.800 3.0485
0.900 3.4351
1.00 4.1144


Change in Elevation
.5 kg cart, height of three textbooks (.112 m)
Position (m) Time (s) Trial 1 Time (s) Trial 2 Time (s) Trial 3
0.100 .4390 .4652 .4680
0.200 .6665 .6831 .6953
0.300 .8962 .8211 .8411
0.400 .9964 .9931 .9577
0.500 1.1191 1.0904 1.0603
0.600 1.2101 1.2302 1.2302
0.700 1.3483 1.3410 1.3655
0.800 1.4409 1.4503 1.4853
0.900 1.5374 1.5478 1.5281
1.00 1.6355 1.6115 1.6332

Position (m) Average time (s)
0.100 .4574
0.200 .6816
0.300 .8528
0.400 .9824
0.500 1.0899
0.600 1.2235
0.700 1.3516
0.800 1.4588
0.900 1.5378
1.00 1.6267

Position (m) Average time2
(s2)
0.100 .2092
0.200 .4646
0.300 .7273
0.400 .9651
0.500 1.1879
0.600 1.4970
0.700 1.8268
0.800 2.1281
0.900 2.3648
1.00 2.6462



Evaluation of Data:

Control
This demonstrates that as the time increases, the position increases at an increasing rate.

In this graph we thought it might straighten out the graph if we squared time. In this graph, as time squared increases position increases at a constant rate.
Equation for linearized graph: P=(.2407m/s2)t2

Change in Mass
This demonstrates that as the time increases, the position increases at an increasing rate.

After linearizing in the same way as the Control graph we see that as time squared increases position increases at a constant rate.
Equation for linearized graph: X=(.2550m/s2)t2

Change in Elevation
This demonstrates that as the time increases, the position increases at an increasing rate.
After linearizing in the same way as the Control graph we see that as time squared increases position increases at a constant rate.
Equation for linearized graph: X=(.3943m/s2)t

Conclusion:
In conclusion for a cart going down a ramp time squared is proportional to the position. The proof of this statement is when tested the cart going down the ramp for each specific situation we were face with a curve opening upward. When the time value was squared the graph became linear. Compared to the position vs. time test, our first experiment and our control, each one of our variables, mass and elevation, changed the slope/ half the acceleration for a greater magnitude than the control. Mass had the smallest increase in magnitude from the control with a difference of only .014m/s2. Elevation had a larger change in its magnitude from the control with a difference of .154 m/s2. Elevation has a larger affect on the velocity of a cart on a ramp than mass.
The graph of position vs. time was graphed and showed a slight curve. We then linearized the graph by squaring the time values to get points that would correspond with a line so an equation could be found. The mathematical equation obtained from the first part of the lab (position vs time control) is: X=(.241m/s 2 )t, where x is the position, t is the time, and .241 is the velocity or half the acceleration per second. The original equation had a y intercept. The y-intercept (0,0) is just a placeholder. It shows that before the timer started the cart had not moved at all, meaning that at 0 seconds the cart’s position would be 0, which is why the intercept was dropped in the final equation. The cart having an intercept can be attributed to lab errors.
The graph of position vs. time with a change in mass was shown on a graph and had a slight curve. To fix this we linearized this by squaring the time values, the t values. The mathematical equation obtained from the (position vs. time with a change in mass): X=(.255m/s 2 )t where X is the position, t is the time, and .255 is the velocity or half the acceleration per second. The original equation had a y intercept. The y-intercept (0, 0) is just a placeholder. It shows that before the timer started the cart had not moved at all, meaning that at 0 seconds the cart’s position would be 0, which is why the intercept was dropped in the final equation.
The graph of position vs. time with a change in elevation was graphed and showed a slight curve. We then linearized the graph by squaring the time values, the t values to get points that would correspond with a line so an equation could be found. The mathematical equation obtained from the first part of the lab (position vs. time with a change in elevation) is: X=(.395m/s2)t where X is the position, t is the time, and .395 is the velocity or half the acceleration per second. The original equation had a y intercept. The y-intercept (0, 0) is just a placeholder. It shows that before the timer started the cart had not moved at all, meaning that at 0 seconds the cart’s position would be 0, which is why the intercept was dropped in the final equation.
Our general equation for all three experiments is X=(1/2a )t In this equation X stands for position, a stands for acceleration, 1/2a stands for the slope of the positon vs. time graph, and t for the time. This equation is correct for all three experiments because no matter what the variable tested is the slope, or1/2a, multiplied by the time as stated in the equation will equal different values dependant upon the slope, or 1/2a that we get which changes when we change what we are testing. In this case affect of mass, and affect of elevation on a cart on a ramp.
There were errors in our lab. Sometimes we accidentally tripped the photogate so that it started when the cart was not released. Of course, we always would reset the timer (photogates) if this happened. Another error is having a y intercept in our line of best fit that shows that before the timer started the cart had is already starting at a positive position which is not true, this is why the intercept was dropped in the final equations.
The position of an item is where an object is during one moment with no movement in space (one point required), while an object’s distance is the amount of space between two areas it has occupied (two points required). Velocity is the rate of change/ ratio of position to time in a direction. Acceleration is the rate of change/ ratio of velocity over time in a direction.

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