Question

You are asked to design a spring that will give a 1180kg satellite a speed of 2.90m/s relative to an orbiting space shuttle. Your spring is to give the satellite a maximum acceleration of 5.00g. The spring's mass, the recoil kinetic energy of the shuttle, and changes in gravitational potential energy will all be negligible.

1- What must the force constant of the spring be?

2- What distance must the spring be compressed?

Answer 1

<< What must the force constant of the spring be? >>

Kinetic energy of satellite = Energy from the spring

(1/2)(m)(V^2) = (1/2)kx^2

where

m = mass of the satellite = 1180 kg (given)
V = velocity of satellite = 2.90 m/sec (given)
k = spring constant
x = length at which spring needs to be compressed

Substituting values,

(1/2)(1180)(2.90^2) =(1/2)(k)(x^2)

kx^2 = 9923.8 --- call this Equation 1

Using Newton's 2nd Law of Motion,

F = ma

kx = ma

where

a = acceleration of the satellite

and all the terms have been previously defined.

and substituting values,

kx = (1180)(5 * 9.8) = 57820

and solving for "x"

x = 57820 /k ---call this Equation 2

Substituting Equation 2 in Equation 1, you will have

k(57820 /k)^2 = 9923.8

Simplifying the above,

(57820)^2/k = 9923.8

k = (57820)^2/9923.8

k = 336882.283 N/m

<< What distance must the spring be compressed? >>

x = 57820 /336882.283 = 0.171623 m