Cart Ramp Lab

Following the ball rolling down the ramp lab, there are numerous questions about what roll different balls, different ramp angles and different starting positions of the ball in relation to the acceleration. In order to answer these questions, the cart ramp lab is used. Here we seek to look at one variable, angle of the ramp.

The primary objective is to look at the relationship between the angle of motion and the acceleration of gravity. The second objective is to look at a quadratic model d = (1/2) a t^2 +v_it + d_i or in vertex form d = (1/2) a (t - h)^2 + k.

This page will walk you through the collection and demonstrate how to analyze the data.

Materials

1 Pasco cart with picket fence and ramp (any cart with a picket fence can be used).
1 Photogate
1 CBL
1 TI-8? calculator

Data Collection

The students are grouped in pairs and assigned a specific height to raise one end of the ramp. Each group uses the same system. Students record the height the ramp is raised in meters and the length of the ramp.

Students move the ramp close to a specified height and them accurate measure the height the ramp has been raised.

Students were instructed to place the photogate 30 cm from the bottom end of the ramp and release the cart 80 cm from the bottom of the ramp. In this manner, we have done our best to control all variables except for the angle of the ramp.

Students line up the cart at the 80 cm mark on the ramp for release. The photogate has been placed at the 30 cm mark.

Use the program FENCE1CM when using the 1 cm photogate or FENCE5CM if using the 5 cm photogate to collect the data. The programs are available as fence.83g . Instructions on downloading or using the program. The program LINKCK must also be on the calculator.

Note: The origin (zero point) of this lab is the photogate with distance measured down the ramp considered to be positive since the distance is recorded on the picket fence passing through the photogate.

Run the appropriate FENCE program for the size picket fence to be used. Place the cart at the desired position, follow the instructions on the screen. Press ENTER when the experiment is ready. The calculator should display READY.... and the cart can be released. As the cart rolls through the photogate, the time and distance is collected and stored in L1 & L2. The graph is displayed. Pressing ENTER a final time will show the quadratic regression y = ax^2+bx+c values for a, b, and c. Note that the acceleration is really 2 times the calculators a value as the model is y = (1/2) acceleration * t^2. The value for the acceleration is also shown.

Students check their data to make sure they have a good graph after running the cart down the ramp.

Analyzing the Individual Group Data

The data for a ramp 1.219 cm in length with one end raised to 0.770 cm is shown in the table below.

Time (sec)	Distance (m)
0	0
0.0039	0.01
0.0078	0.02
0.0116	0.03
0.0154	0.04
0.0191	0.05
0.0228	0.06
0.0265	0.07
0.0301	0.08
0.0337	0.09
0.0373	0.1
0.0409	0.11
0.0444	0.12

When the data is entered into L1 and L2 with a stat plot set up on these lists, a graph similar to the the one at the right is observed.
	Students look at this graph and think the graph is linear. If a linear regression is performed [Note: STAT - CALC - LinReg(ax+b) L1,L2 ], the calculator returns a linear function as shown below.
Note that the regression coefficient shows a reasonably good fit. If you graph the line on the data, it also appears to fit well. If you want to automatically enter the regression into a function graph, add a ",Y1" to the end to the regression request. Hence, the line will appear as STAT CALC LinReg(ax+b) L1,L2,Y1. Function graph values (Y1 etc) are found in the VARS, Y-Vars, Function menu.
	However, as noted in the previous lab, objects moving down an inclined ramp tend to accelerate and the linear model assumes that the velocity is constant. To note acceleration, we need to use a quadratic model. To find a quadratric regression, enter STAT CALC LinReg(ax+b) L1,L2,Y1 (the Y1 will enter the quadratic regression into the Y1 function graph. The quadratic regression for the data shown in the table above is shown at the left.
Note the difference in the R^2 correlation coefficient from the linear regression. This graph shown below fits the data better than the previous linear equation.

Interpreting the Individual Group Data

The model d = 3.33 t^2 + 2.56t - 6.59E-5 is shown by the calculator as the regression equation for the data. The value 3.33 m/s^2 is one-half of the acceleration. Hence, the acceleration is 2*3.33 m/s^2 or 6.66 m/s^2. The "b" value is the initial velocity when time is zero. Hence, the model shows that the velocity of the cart was traveling at the rate of 2.56 m/s when it triggered the photogate. Finally, the initial distance when time was zero is -6.59E-5 which means -6.59 X 10^-5 or another words zero meters which is what we would expect because the cart is at the photogate when the timing mechanism is triggered.

If one mathematically solves for the vertex from a quadratic equation y = ax^2 + bx + c using a method called completing the square, we find that the vertex occurs at ( h, k) where h = -b/(2a) and k = c - b^2/(4a) . Hence the time of the vertex is -b/(2a) and the distance the vertex occurs from the origin is c - b^2/(4a).

In this case h=-2.56/(2*3.33)=-0.38 seconds. The k value of the vertex is 0-(2.56^2)/(4*3.33) or -0.49 meters. This means that the cart had a velocity of zero at the vertex which occurred 0.38 seconds before the cart passed through the photogate and the distance to the vertex is 0.49 meters up the ramp. (Recall that down the ramp was set as a positive distance). By the way, where was the cart released? Answer: (80 cm.) Where was the photogate? Answer (30 cm) What is the difference? Answer (50 cm). The math really does match with the science!

Collecting a Classroom Set of Data

To record the classroom set of data, you will need to discuss how to record the angle. This will somewhat vary with the math background of your students. Perhaps the easiest way to look at the problem is tto take to boards which are different lengths and raise one end of each the same distance above the table. What helps determine the angle? Of course it is the length of the board. Hence, let's divide the height raised by the length of the board. Mathematically, that is y/r which is equal to the sine of the angle. Inserting this value into the inverse sine function on a calculator will display the angle. (Beware: The calculator may be set to radian or degree mode.) Let's have the students record the ratio of height/length (sine of the angle) of ramp and their acceleration. The table below shows the data from each group.

Graphing the data yields the following:

Since the graph appears to be linear, running a linear regression yields the following:

The graph of the linear model on the data appears as:

Interpreting the data, the x-axis is the ratio of the height divided by the ramp length. This could also be known as the sine of the angle. The y -axis is the acceleration in meters per seconds squared. Hence the model would be acceleration (m/s^2) = 9.45 (m/s^2)* sin(angle) + 0.02 m/s^2. Since the y-intercept is 0.3% of the largest value and we would expect zero acceleration when the ramp is at an angle of zero, we will call the y-intercept zero and the model is:

One would expect a value of 9.8 m/s^2 for the slope of the line. Hence the percent error in the lab is 3.6%.

Click here for an updated explanation using Logger Pro software.

If you are an educator or interested in this activity and wish to have more information, send an e-mail to jrynear@lps.org or jwelker@lps.org.