An infinite horizontal string is plucked. The vertical displacement of the string is the solution to the following initial value problem x E R, t > 0 u(x, 0) = $(x) ди (x, 0) = 0 with 1- |æ|/2, for |x| < 2 $(x) = for |x| > 2 (i) Using d'Alembert's solution, give an explicit solution to this initial value problem.

Range = ______ a) 2 vi2 sin(θ) cos (θ) / g b) vi2 sin(θ) cos (θ) / g c) 2 vi sin(θ) cos (θ) / g d) vi sin(θ) cos (θ) / g

The displacement x(t) cms at time t seconds, of a spring moving along a line is given by the formula: x(t) = 8e- cos(2t). %3D Compute the velocity x'(t) and acceleration x"(t) of the spring and show that these are related by a formula of the kind x" + ax' + bx = 0 for suitable constants a and b which must be %3D found. Also estimate the maximum speed of the particle and its maximum acceleration, during its motion, with explanation. Also discuss the behavior of the motion for large times.

Find dw/dt using the appropriate Chain Rule. Function w = x² + y² x = 3t, y = 2t dw dt Value t = 2 Evaluate dw/dt at the given value of t.