A ball is dropped from the top of a tall building. If air resistance is taken into account, recall that the downward velocity v of the ball is modeled by the differential equation: d v / d t = g − p vwhere g = 9.8 m / s e c 2 is the acceleration due to gravity, and p = 0.1 is the drag coefficient. Assuming the initial velocity of the ball is zero, use Euler’s method with a step size of h = .25 sec to estimate the velocity of the ball after one second. Keep track of three decimal places during the calculation.I just started learning the Euler's method. Can someone help me with this problem. I know that: d v / d t = f ( v , t ) = g − p vI plug the numbers given into the equation: d v / d t = 9.8 − .25 vchange of x = step size (h) = .25I think ( v ( 0 ) , t ( 0 ) ) = ( 0 , 0 )??

Question

A ball is dropped from the top of a tall building. If air resistance is taken into account, recall that the downward velocity v of the ball is modeled by the differential equation:  d  v      /    d  t  =  g  −  p  vwhere g = 9.8   m      /    s  e      c    2   is the acceleration due to gravity, and p = 0.1 is the drag coefficient. Assuming the initial velocity of the ball is zero, use Euler’s method with a step size of h = .25 sec to estimate the velocity of the ball after one second. Keep track of three decimal places during the calculation.I just started learning the Euler&#039;s method. Can someone help me with this problem. I know that:  d  v      /    d  t  =  f  (  v  ,  t  )  =  g  −  p  vI plug the numbers given into the equation:   d  v      /    d  t  =  9.8  −  .25  vchange of x = step size (h) = .25I think   (  v  (  0  )  ,  t  (  0  )  )  =  (  0  ,  0  )??

Brooks Hogan · Accepted Answer

First, we discretize our domain into points       t    n    =  n  h  . Then, Euler&#039;s method is given by      V          n      +      1        =      V    n    +  h  f  (      V    n    )  ,corresponding to the equation      v    ′    =  f  (  v  )  .The sequence   (      V    n    ) should approximate the solution in the sense that       V    n    ≈  v  (      t    n    ). In your case, we find that   f  (  v  )  =  g  −  p  v  , and since we want to compute up to   t  =  1 with   h  =  0.25  , we want to compute 4 iterations (this will correspond to time-stepping up to one second). Since       V    0   is just the initial condition, we find                              V          0                                    =        0                                      V          1                                    =                  V          0                +        h        f        (                  V          0                )        =        h        (        g        −        p                  V          0                )                                      V          2                                    =                  V          1                +        h        f        (                  V          1                )        =                  V          1                +        h        (        g        −        p                  V          1                )                                      V          3                                    =                  V          2                +        h        f        (                  V          2                )        =                  V          2                +        h        (        g        −        p                  V          2                )                                      V          4                                    =                  V          3                +        h        f        (                  V          3                )        =                  V          3                +        h        (        g        −        p                  V          3                )            You can calculate these directly. In particular, the final one gives you      V    4    ≈  v  (      t    4    )  =  v  (  4  h  )  =  v  (  1  )  .

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