Answer:
x=0.46m, speed=7.9m/s
Explanation:
Using the concept of conservation of energy:
1. kinetic energy of mass m and velocity v: [tex]E_k=\frac{1}{2}mv^2[/tex]
2. gravitational potential energy of mass m, grav. acc. g and height h: [tex]E_g=mgh[/tex]
3. potential energy in a spring with spring constant k and displacement from equilibrium x: [tex]E_s=\frac{1}{2}kx^2[/tex]
Calculating x:
[tex]\frac{1}{2}mv_a^2=\frac{1}{2}kx^2[/tex]
[tex]x=\sqrt{\frac{m}{k}}v_a[/tex]
Calculating the speed:
[tex]\frac{1}{2}mv_a^2 +mgh_a=\frac{1}{2}mv_b^2+mgh_b + W_{friction}[/tex]
[tex]h_a=0, h_b=2R,W_{friction}=F_{friction}\times distance=7\pi R[/tex]
[tex]\frac{1}{2}mv_a^2=\frac{1}{2}mv_b^2+2mgR+7\pi R[/tex]
Solving for [tex]v_b[/tex]:
[tex]v_b=\sqrt{v_a^2-4gR-14\pi\frac{R}{m}}[/tex]
