Elastic Potential Energy — Pogo Stick (Worked Example)

Medium

Problem

A pogo stick spring has k = 8000 N/m. At the bottom of a bounce it compresses 0.15 m. How much elastic potential energy is stored at that instant?

Given

k = 8000 N/m
x = 0.15 m

Find

The elastic potential energy PE.

Principle

At maximum compression the jumper momentarily stops — all the bounce energy sits in the spring as ½kx².

Formula

PE = \frac{1}{2}kx^2

PE = ½kx²

Solution

Write the formula: $PE = \frac{1}{2}kx^2$
Substitute: $PE = \frac{1}{2}(8000)(0.15)^2$
Calculate: $PE = 4000 \times 0.0225 = 90\ \mathrm{J}$

90 J

Unit check

N/m × m² = J — energy, correct.

What this means physically

90 J — enough to lift the 40 kg rider about 23 cm. Real pogo sticks return most of this as upward kinetic energy, which is why each bounce needs only a small extra push.

Common mistake

Squaring k instead of x, or forgetting the ½. The x² dependence means doubling the compression quadruples the stored energy.