Question

You are asked to design a spring that will give a 1080 kg satellite a speed of 2.65 m/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.

A)What must the force constant of the spring be?

Take the free fall acceleration to be g = 9.80 m/s2 .

b)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 = 1080 kg (given)
V = velocity of satellite = 2.65m/sec (given)
k = spring constant
x = length at which spring needs to be compressed

Substituting values,

(1/2)(1080)(2.65^2) =(1/2)(k)(x^2)

kx^2 = 7584.3 --- 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 = (1080)(5 * 9.8) = 52920

and solving for "x"

x = 52920/k --- this Equation 2

Substituting Equation 2 in Equation 1, you will have

k(52920/k)^2 = 7584.3

Simplifying the above,

(52920)^2/k = 7584.3

k = (52920)^2/7584.3

k = 369253 N/m

What distance must the spring be compressed?

either Equation 1 or Equation 2 to solve for this

Equation 2...

X =   52920/369253 = 0.1433m