Pogo Stick Problem: Steven's Speed & Max Compression

[bleedblue1234]

Homework Statement

Steven, with a mass of 47.8 kilograms, jumps up to a height of 1.26 meters on a pogo stick with a spring constant of 14700 Newtons per meter. (The bottom of the uncompressed pogo stick spring is 1.26 meters above the ground.)

a. What is Steven's speed when the spring is compressed 22.0 centimeters? Hint: When the spring is compressed 22.0 centimeters, Steven has fallen more than just 1.26 meters.

b. What is the maximum distance the spring compresses?

Homework Equations

The Attempt at a Solution

 

[jbsteele]

a. Since the spring is compressed 22.0 cm, Steven has fallen 1.26m - 0.22m = 1.04m. Using conservation of energy, we can solve for the speed when the spring is compressed 22.0 cm. PE(initial) = PE(final) + KE(final)mgh = (1/2)kx^2 + (1/2)mv^2 v = sqrt[2gh/k] v = sqrt[2(9.8m/s^2)(1.04m)/(14700N/m)] v = 0.60 m/s

 

[tomcue]

a. To calculate Steven's speed when the spring is compressed 22.0 centimeters, we can use the equation for conservation of energy:

mgh = 1/2kx^2 + 1/2mv^2

Where m is the mass of Steven, g is the acceleration due to gravity (9.8 m/s^2), h is the initial height (1.26 meters), k is the spring constant, x is the distance the spring is compressed (0.22 meters), and v is the final velocity.

Substituting in the given values, we get:

47.8 kg * 9.8 m/s^2 * 1.26 m = 1/2 * 14700 N/m * (0.22 m)^2 + 1/2 * 47.8 kg * v^2

Solving for v, we get:

v = 4.19 m/s

Therefore, Steven's speed when the spring is compressed 22.0 centimeters is 4.19 m/s.

b. To find the maximum distance the spring compresses, we can use the equation for maximum compression of a spring:

x = mg/k

Where m is the mass of Steven, g is the acceleration due to gravity (9.8 m/s^2), and k is the spring constant.

Substituting in the given values, we get:

x = 47.8 kg * 9.8 m/s^2 / 14700 N/m

Solving for x, we get:

x = 0.032 meters

Therefore, the maximum distance the spring compresses is 0.032 meters.