Numerical Differentiation: Problems
- Compare the basic formulas and the high-accuracy formulas to calculate the first and second derivatives of at with . Clearly indicate the order of the error term in the approximation used. Calculate the relative error in each case.
A plane is being tracked by radar, and data are taken every two seconds in polar coordinates and .
Time (sec.) 200 202 204 206 208 210 (rad) 0.75 0.72 0.70 0.68 0.67 0.66 (m.) 5120 5370 5560 5800 6030 6240
The vector expression in polar coordinates for the velocity and the acceleration are given by:
Use the centred finite difference basic formulas to find the velocity and acceleration vectors at sec. as a function of the unit vectors and . Then, using the norm of the velocity and acceleration vectors describe the plane speed and the magnitude of its acceleration.
- Fick’s first diffusion law states that:
where is the mass transported per unit area and per unit time with units , is a constant called the diffusion coefficient given in , is the concentration given in and is the distance in . The law states that the mass transported per unit area and per unit time is directly proportional to the gradient of the concentration (with a negative constant of proportionality). Stated differently, it means that the molecules tend to move towards the area of less concentration. If the concentration of a pollutant in the pore waters of sediments underlying the lake is constant throughout the year and is measured to be:
0 1 3 , 0.06 0.32 0.6
where is the interface between the lake and the sediment. By fitting a parabola to the data, estimate at . If the area of interface between the lake and the sediment is , and if , how much pollutant in would be transported into the lake over a period of one year?
- The positions in m. of the left and right feet of two squash players (Ramy Ashour and Cameron Pilley) during an 8 second segment of their match in the North American Open in February 2013 are given in the excel sheet provided here. Using a centred finite difference scheme, compare the average and top speed and acceleration between the two players during that segment of the match. Similarly, using a parabolic spline interpolation function, compare the average and top speed and acceleration between the two players. Finally, compare with the average speed of walking for humans.