Project: Study of motion of a falling ball using first and second derivatives
The positions and velocities of a falling ball at time intervals of 0.02 second are given in the table below:
Time (seconds) / Height (meters) / Velocity (meters/second)0.00 / 0.290864 / -0.16405
0.02 / 0.284279 / -0.32857
0.04 / 0.274400 / -0.49403
0.06 / 0.260131 / -0.71322
0.08 / 0.241472 / -0.93309
0.10 / 0.219520 / -1.09409
0.12 / 0.189885 / -1.47655
0.14 / 0.160250 / -1.47891
0.16 / 0.126224 / -1.69994
0.18 / 0.086711 / -1.96997
0.20 / 0.045002 / -2.07747
0.22 / 0.000000 / -2.25010
Question 1. Use Maple to enter the three columns of data, and then use the scatter plot command to make a graph of Time versus Height and a graph of Time versus Velocity. Examine the graphs. What model (function) seems to give the best fit for the scatter plot of heights? What type of model (function) seems to give the best fit for the scatter plot of velocities of the falling ball? Describe any relationships that you observe between the two graphs.
Question 2. A model for the position function of a falling body (in the absence of air resistance which acts like a friction to retard the fall) is given by
s(t)=-1/2gt2+v0+s0
where g is the acceleration due to gravity and v0 and s0 are the initial position of the ball and the initial velocity imparted to the body respectively. Use the Maple least squares command to find the best quadratic model for the Time (x) versus Height (y) data. Use your result to determine the ball’s initial height and initial velocity, the velocity function and the acceleration function. Record your results below:
Position function: s(t)=
Initial Height: s0=
Initial Velocity: v0=
Velocity Function: v(t)=s`(t)=
Acceleration Function: a(t)=s``(t)=
Question 3. Use Maple to plot the Time versus Velocity data and the velocity function that you determined in Question 2 on the same axes. Is there a good fit between the data and the function? Why or why not? Explain any discrepancies.
Question 4. A model for the velocity function has the form
v(t)=-gt+v0
where again g is acceleration due to gravity and v0 is the initial velocity of the falling body. Use the least squares method to find the best linear fit for the Time versus Velocity data. Use this velocity function to determine the acceleration function, and record your results below:
Velocity Function: v(t)
Acceleration Function: a(t)=v`(t)
Question 5. Use Maple to plot the Time versus Velocity data along with the velocity functions from Questions 2 and 4 on a single pair of axes. Which velocity function gives the better fit to the data? Explain why these two velocity functions that describe the same data are different.
Question 6. Of the two acceleration functions that you found in Questions 2 and 4, which give the closer estimate to the actual value of the acceleration due to gravity, -9.8 meters/sec2? Calculate the percent error in the two cases using
Percent error=Abs(Expected value –Observed value)/Expected value x100%
Do you think that these are reasonable estimates of g? Explain.
Question 7. Use your position function from Question 2, and the Maple solve command to find the time when the ball hits the ground. How does this value compare to the observed value of 0.22 seconds. Calculate the percent error as in Question 6.
A second ball falls according to the data below:
Time (seconds) / Height (meters) / Time (seconds) / Height (meters)0.00 / 0.806736 / 0.32 / 1.149180
0.02 / 0.857225 / 0.34 / 1.141500
0.04 / 0.904422 / 0.36 / 1.126130
0.06 / 0.946131 / 0.38 / 1.105280
0.08 / 0.985644 / 0.40 / 1.082230
0.10 / 1.020760 / 0.42 / 1.056980
0.12 / 1.052590 / 0.44 / 1.026250
0.14 / 1.080030 / 0.46 / 0.992230
0.16 / 1.103080 / 0.48 / 0.954912
0.18 / 1.122840 / 0.50 / 0.913203
0.20 / 1.137110 / 0.52 / 0.868201
0.22 / 1.149180 / 0.54 / 0.819907
0.24 / 1.156870 / 0.56 / 0.767222
0.26 / 1.160160 / 0.58 / 0.711244
0.28 / 1.161260 / 0.60 / 0.651974
0.20 / 1.156870 / 0.62 / 0.589411
Question 8. Use Maple to find the best-fit position function determined by the data. Use this position function to find the initial height and initial velocity. Do you think that the ball was dropped or thrown? Calculate the velocity function. Find the time when the ball will hit the ground and the velocity of the ball at that time, often referred to as terminal velocity.
Position Function: s(t)=
Initial Height: s0=
Initial Velocity: v0=
Velocity Function: s`(t)=v(t)=
Question 9. For which t values is s`(t) positive? For which t values is s`(t) negative. Explain what the graph of v(t)=s`(t) tells you about the graph of s(t). What does the graph of a(t)=s``(t) tell you about the graph of s`(t)? What does the graph of a(t) tell you about the graph of s(t)?
Question 10. Choose four distinct points (reasonably separated) in the domain of s(t), and use maple to plot a graph showing the tangent lines to the graph of s(t) at the chosen points. For what value of t is the tangent line horizontal? What is the value of v(t) at this time?