A child's pogo stick(figure) stores energy in a spring with a force constant of `2.5xx10^4Nm`. At position (A) `(x_A=0.10m)`, the spring compression is a maximum and the child is momentarily at rest. At position (B) `(x_B=0)`, the spring is relaxed and the child is moving upward. At position (C), the child is again momentarily at rest at the top of the jump. The combined mass of child and pogo stick is `25kg`. (a) Calculate the total energy of the child-stick-earth system if both gravitational and elastic potential energies are zero for `x=0`. (b) Determine `x_C`. (c) Calculate the speed of the child at `x=0`. (d) Determine the value of x for which the kinetic energy of the system is a maximum. (e) Calculate the child's maximum upward speed.

Explanation

`k=2.5xx10^4Nm^-1`, `m=25kg`, `X_A=-0.10m`,
`U_g:|_(x=0)=U_s:|_(x=0)=0`
a. `E_(mech)=K_A+U_(gA)+U_(sA)=0+mgx_A+1/2kx_A^2`
`=25xx10xx(-0.100)+1/2xx2.5xx10^4xx(-0.100)^2`
`=-25J+125J=100J`
b. Since only conservation forces are involved, the total energy of the child-pogo stick-earth system at point C is the same as that at point A.
`K_C+U_(gC)+U_(sC)=K_A+U_(gA)+U_(sA)`
`0=25xx10xxx_C+0=0-25J+125J`
`x_C=0.40m`
c. `K_B+U_(gBB)+U_(sB)=K_A+U_(gA)+U_(sA)`
`1/2xx25xxv_B^2+0+0=0+(-25)+125`
`v_B=2sqrt2ms^-1`
d. K and v are at a maximum when `a=sum F//m=0` (i.e., when the magnitude of the upward spring force equals the magnitude of the downward gravitational force).
This occurs at `xlt0`, where `k|x|=mg`.
`|x|=(25xx10)/(2.5xx10^4)=10xx10^-3m=10mm`
Thus, `K=K_(max)` at `x=-10mm`
e. `1/2xx25xxv_(max)^2=25xx10xx[(-0.100)-(-0.01)]+1/2(25xx10^4)[(-0.100)^2-(-0.01)^2]`
`impliesv_(max)=2.85ms^-1`