Two objects A and B (both masses are m) are connected by a spring of elastic constant k and placed on a frictionless surface. At some point, A gains an initial velocity v parallel to the spring, and this system will thus start moving. Knowing that the spring is undeformed in the beginning, determine (a) the highest elastic potential energy that can be stored in the spring and (b) the highest velocity that can be reached by B.

a)   Em_f = ½ m vₐ²,  b)    v = vₐ /√2

Explanation:

a) For this part we use conservation of energy

   starting point .. block A moving, spring unstretched

          Em₀ = K = ½ m vₐ²

end point. Stretched spring

          Em_f = K_e = ½ k x²

energy is conserved

          Em₀ = Em_f

          Em_f = ½ m vₐ²

b) Let's analyze the movement a little, block A began to move at a speed va, I stretch the spring an amount Dx, it exerts a force on block b that begins to move and the elongation of the spring decreases.

In all this process there is no friction force, therefore the energy is conserved, therefore the maximum energy stored in the spring must be distributed among the bodies.

           Em₀ = K_e = E₀

where E₀ is the initial energy of block a

            E₀ = ½ m vₐ²

At the point where we are in equilibrium

          Em_f = Kₐ + K_b = ½ m vₐ² +1/2 m

so that the spring does not stretch or shrink, the two bodies must go at the same speed.

          Em_f = m v²

energy is conserved

          Em₀ = Em_f

          ½ m vₐ² = m v²

           v = vₐ /√2

therefore both blocks must go at this speed

i think the answer is a, hope this : )