Problems for
Intermediate Methods in Theoretical Physics
Edward F. Redish
Solving N2 by stepping
Consider dropping a ball at time t = 0 at the location y = 0. Take the y-axis as being vertical with positive upward. Ignore air resistance and the curvature of the earth.
Write the exact solutions for y(t) and v(t) at the time t_{n} = n Δt.
Write the Euler approximation for N2 as a stepping rule for getting y_{n+1} = y(t_{n+1}) and v_{n+1} = v(t_{n+1}) if you know y_{n} and v_{n}.
Starting from (y_{0}, v_{0}) = (0, 0), use the Euler rule to find the first three steps, (y_{1}, v_{1}), (y_{2}, v_{2}), and (y_{3}, v_{3}). Can you generalize your result and make a conjecture for the answer for the n-th step? Can you prove that this is correct? (Hint: Try substituting your conjecture into the equation for (b).)
How bad is the answer? Does it get better at time gets longer? or worse? Discuss why you think it does what it does.
RETURNS
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Edward F. Redish
Department of Physics
University of Maryland
College Park, MD 20742
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Email: redish@umd.edu