pogo stick question regarding potential and kinetic energy.?

Question

a childs pogo stick stores energy in a spring with spring constant 21800 N/M. at position A(x1=-.06m) the spring compression is a maximum and the child is momentarily at rest. at position B ( x=0 m) the spring is relaxed and the child is moving upward. at position C (x2= height) the spring is still relaxed and the child is again momentarly at rest at the top of the jump. the acceleration of gravity is 9.8 m/s^2. mass of child and spring = 29 kg. at position C determine the height ?at position B calculate the speed of the child at x=0 ? determine the magnitude of x for which the kinetic energy of the system is at a max ?

Answer 1

When the stick is compressed, the energy stored in the stick is
.5*k*x^2

when the stick is completely uncompressed the energy has been converted to kinetic energy and potential energy.

.5*m*v^2+m*g*h
where h=x+.06
Spring energy=kinetic+potential

When the child reaches apogee, all of the energy is potential

.5*k*x^2=m*g*h

At position C
x=-.06+(21800*.06*.06)/
(2*29*9.8)

x=.078 m

the speed at position b

.5*k*x^2=.5*m*v^2+m*g*.06

21800*.06*.06-2*29*9.8*.06=
29*v^2

v=sqrt((21800*.06*.06-2*29*9.8*.06)/
29)

=1.24 m/s



The maximum kinetic energy may not be at position b, it may be for
-0.6
the kinetic energy will be at maximum when the square of the velocity is at maximum

This requires a bit of care to express KE as a function of x:


I will translate my x coordinate to be x=0 when the spring is fully compressed and stores energy equal to .5*k*.06^2

The incremental energy stored in the spring at point x is
.5*k*(.06-x)^2

The potential energy gained at point x is
m*g*x

KE(x)=.5*k*.06^2-
.5*k*(.06-x)^2-
m*g*x
for 0<=x<=+.06
take the first derivative

dKE/dx=-k*x+.06*k-m*g

set equal to zero and solve for x

x=(.06*k-m*g)/k

plugging in the numbers
x=.047 m

to translate back to the original coordinate system where

.047-.06
=-.013 m

I believe this can be solved without calculus. As the spring decompresses the force it has decreases. At some point the force of the spring will be less than the weight of the child and stick and the upward acceleration will end, which will be the point when the kinetic energy is at the maximum

To find this point of extension:

k*x=m*g
x=m*g/k
=29*9.8/21800
=.013
In the original coordinate system, this will be the extension from zero negative

-.013 m
same answer as using calculus but much simpler.
The advantage of using calculus, though, is that the answer can be tested.


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