Amusement Park Physics

Evaluating Data from the Calculator Based Laboratory (CBL)

The data below was collected as described on the previous page. The next question, is how can one analyze the data. First, we will restrict our analysis of the data to the initial fall from a height and rebound back up the tower until the ride starts to fall again. This section of data is shown in the graph below. Furthermore, we will not use calculus techniques, so the results are at best an approximation.

Acceleration (m/sec^2) vs. Time

The manipulation of the data is quite complex, but the goal is to arrive at a distance versus time graph of the ride based on the acceleration. Normally, one can approximate the velocity from a distance vs. time graph by taking the change in distance and dividing it by the change in time. Repeating this process on the velocity vs time graph, one can take the change in velocity and divide it by the change in time which is acceleration. We will reverse this process to return to distance from acceleration. To help simplify the situation, assume that the following data has been gathered.

Time	Accel
0.0	9.8
0.1	10.8
0.2	13.8
0.3	17.8
0.4	25.8
0.5	29.8

The get velocity vs. time from acceleration, one needs to:

1. Find the net acceleration. The data was collected with the acceleration due to gravity included. Find the acceleration due to the ride. By taking the acceleration data and subtracting 9.8 m/s^2, the net acceleration or acceleration created by the ride is calculated.

   Calculations			Formula
   Time	Accel	Net Accel	Time	Accel	Net Accel
   0.0	9.8	0.0	T1	A1	=A1-9.8
   0.1	10.8	1.0	T2	A2	=A2-9.8
   0.2	13.8	4.0	T3	A3	=A3-9.8
   0.3	17.8	8.0	T4	A4	=A4-9.8
   0.4	25.8	16.0	T5	A5	=A5-9.8
   0.5	29.8	20.0	T6	A6	=A6-9.8

2. Find the change in velocity. The change in velocity is acceleration multiplied by the change in time. We will use the average the two time values and calculate the velocity.

   Calculations				Formula
   Time	Net Accel	Avg Time	Delta Velocity	Time	Net Accel	Avg Time	Net Accel
   0.0	0.0	0.05	0.1	T1	A1	=(T1+T2)/2	=(T2-T1)*A2
   0.1	1.0	0.15	0.4	T2	A2	=(T2+T3)/2	=(T3-T2)*A3
   0.2	4.0	0.25	0.8	T3	A3	=(T3+T4)/2	=(T3-T2)*A3
   0.3	8.0	0.35	1.6	T4	A4	=(T4+T5)/2	=(T4-T3)*A4
   0.4	16.0	0.45	2.0	T5	A5	=(T5+T6)/2	=(T5-T4)*A5
   0.5	20.0			T6	A6

3. The velocity shown is the change in velocity. We need to have the total velocity at that time. In other words, the velocity at time t is the sum of the change in velocity up to time t.

   Calculations			Formula
   Avg Time	Delta Velocity	Velocity	Avg Time	Delta Velocity	Velocity
   0.05	0.1	0.1	T1	DV1	=DV1
   0.15	0.4	0.5	T2	DV2	=DV1+DV2
   0.25	0.8	1.3	T3	DV3	=DV1+DV2+DV3
   0.35	1.6	2.9	T4	DV4	=DV1+DV2+DV3+DV4
   0.45	2.0	4.9	T5	DV5	=DV1+DV2+DV3+DV4+DV5

4. Now that we know the velocity at time t, we will repeat the process and find the change in distance which is the change in time multiplied by the velocity. Once again, average the time and find the change in displacement by multiplying the velocity by the change in time.

   Calculations				Formula
   Avg Time	Velocity	Avg Time	Delta Displacement	Avg Time	Velocity	Avg Time	Delta Displacement
   0.05	0.1	0.1	0.05	T1	V1	=(T1+T2)/2	=(T2-T1)*V2
   0.15	0.5	0.2	0.13	T2	V2	=(T2+T3)/2	=(T3-T2)*V3
   0.25	1.3	0.3	0.29	T3	V3	=(T3+T4)/2	=(T4-T3)*V4
   0.35	2.9	0.4	0.49	T4	V4	=(T4+T5)/2	=(T5-T4)*V5
   0.45	4.9			T5	V5

5. Finally, the displacement at time t is the sum of the displacements up to that time.

   Calculations			Formula
   Time	Delta Displacement	Displacement	Time	Delta Displacement	Displacement
   0.1	0.05	0.05	T1	DD1	=DD1
   0.2	0.13	0.18	T2	DD2	=DD1+DD2
   0.3	0.29	0.47	T3	DD3	=DD1+DD2+DD3
   0.4	0.49	0.96	T4	DD4	=DD1+DD2+DD3+DD4

Repeating the process demonstrated above with the actual data from an amusement park, the calculated velocity vs time graph is shown below. Note that the velocity was determine to be zero when the ride started. At about 8.5 seconds, the velocity was about 20 m/s in the negative direction (toward the ground). At about 10 seconds, the velocity is zero and the acceleration is the greatest. Nearing the 11 second mark, the velocity upward is nearing its maximum of about 14 m/s before the ride starts to slow and the rider as back at the 'top' at about 13.5 seconds and the velocity is once again zero.

The graph below is the calculated height vs time graph. The initial height is given at approximately 94 meters. Therefore, the graph has been shifted so that the height above the ground is displayed with the initial height as given. One will note that the minimum height after the first descent is calculated to be about 35 m when the time is approaching the 10 second mark. After the first rebound, the rider reaches a height of about 67 meters at about 13.5 seconds.