Two identical light springs are each attached to a rigid wall on a smooth horizontal surface. A block of mass m can be connected to the free ends of the springs in two different ways:
Case A: The block is attached to both springs in parallel (both springs side-by-side, their free ends fixed to the same face of the block).
Case B: The block is attached to the springs in series (one spring joined to the block, the other spring joined to the free end of the first spring).
Each individual spring has force constant k.
For both cases the block is pulled towards the wall so that the overall compression of the arrangement is the same distance x and is then released from rest.
(a) Obtain the effective force constant k_eff for Case A and for Case B.
(b) Derive an expression for the maximum speed v_A of the block in Case A and the maximum speed v_B in Case B.
(c) Show analytically which arrangement gives the greater maximum speed for the same values of m, k and x.
(d) If m = 1.0 kg, k = 600 N m⁻¹ and x = 0.05 m, calculate v_A and v_B.