A block of mass `m` moving at a speed `v_0` compresses a spring of spring constant `k` as shown in the figure. Find (a) the maximum compression of spring and (b) speed of the block when the spring is compressed to half of maximum compression.

Explanation

Position(1): `K_1=(1)/(2)mv_0^2`,`U_(g1)=0`,`U_(s1)=0`
(2): `K_2=0`,`U_(g2)=0`,`U_(s1)=(1)/(2)kx_0^2`
(3): `K_3=(1)/(2)mv^2`,`U_(g3)=0`,`U_(s3)=(1)/(2)k((x_0)/(2))^2`
`U_g`: gravitational potential energy
`U_S`: energy stored in spring
By the mechanical energy conservation, we have
`K_1+U_(g1)+U_(s1)=K_3+U_(g2)+U_(s2)`
`(1)/(2)mv_0^2=(1)/(2)kx_0^2impliesx_0=v_0sqrt((m)/(k))`
`x_0`: maximum compression of spring
`K_1+U_(g1)+U_(s1)=K_3+U_(g3)+U_(s3)`
`(1)/(2)mv_0^2=(1)/(2)mv^2+(1)/(2)k((x_0)/(2))^2`
`=(1)/(2)mv^2+(1)/(2)k(v_0^2)/(4)xx(m)/(K)`
`(1)/(2)mv^2=(3)/(8)mv_0^2`
`impliesv=(sqrt3)/(2)v_0`