Question

This problem deals with a mass m on a spring (with constant k) that receives an impulse po = mv, at time t= 0. Show that the following initial value problems have the same solution. Thus the effect of po(t) is to impart to the particle an initial momentum po- mx" + kx = 0, x(0) = 0, x'(0) = V, and mx"' + kx = p. 8(t), x(0) = 0, x'(0) = 0 Click the icon to view the table of Laplace transforms. Let x(t) = xy(t) in the first equation and let L{xy(t)} = X (s). Take the Laplace transform of the first equation. Xy(s) The Laplace transform of 8(t-a) is defined as L{8(t-a)}= Therefore, L L {po 8(t)} = 0 Let x(t) = x2(t) in the second equation and let L{x2(t)} = x2(s). Take the Laplace transform of the second equation. X2(s)=0 Take the difference of Xy(s) and X2(s) and simplify using the fact that po = mvo |X1 (s) – X2(s)|-| Recall that if F(s) = G(s) for all s> (for some c), then f(t) g(t) wherever on [0, +00) both f and g are continuous. Therefore, Xy and Xare identical solutions to the initial value problems, concluding the proof.

Answer 1

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