A child\'s pogo stick as shown below stores energy in a spring with a force cons

Question

A child's pogo stick as shown below stores energy in a spring with a force constant of 2.40 times 10^4 N/m. At position (A)(x_A = -0.150 m), 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 the child and the pogo stick is 32.5 kg. Although the boy must lean forward to remain balanced, the angle is small, so let's assume the pogo stick is vertical. Also assume the boy does not bend his legs during the motion. Calculate the total energy of the child-stick-Earth system, taking both gravitational and elastic potential energies as zero for x = 0. Determine x_c. Calculate the speed of the child at x = 0. Determine the value of x for which the kinetic energy of the system is a maximum. Calculate the child's maximum upward speed.

Explanation / Answer

total energy = initial kinetic energy + initial gravitational potential energy + initial elastic potential energy

total energy = 0 + 32.5 * 9.8 * -0.15 - 0.5 * 2.4 * 10^4 * 0.15^2

total energy = -317.775 J

by conservation of energy

initial energy = final energy

-317.775 = 32.5 * 9.8 * Xc - 0.5 * 2.4 * 10^4 * Xc^2

Xc = 0.176 m

again by conservation of energy

at x = 0 m, potential energy will be 0

-317.775 = 0.5 * mv^2 + 0

-317.775 = 0.5 * 32.5 * v^2

v = 4.42 m/s

speed of child = 4.42 m/s

kinetic energy of the system will be maximum when potential energy is 0 so,

at x = 0 mm the potential energy of the system is 0

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