Answer:
[tex]380697.33\ \text{N/m}[/tex]
[tex]0.138\ \text{m}[/tex]
Explanation:
m = Mass rocket = 1070 kg
v = Velocity of rocket = 3.75 m/s
a = Acceleration of rocket = 5g
g = Acceleration due to gravity = [tex]9.81\ \text{m/s}^2[/tex]
The energy balance of the system is given by
[tex]\dfrac{1}{2}kx^2=\dfrac{1}{2}mv^2\\\Rightarrow kx=\dfrac{mv^2}{x}\\\Rightarrow kx=\dfrac{1070\times 3.75^2}{x}\\\Rightarrow kx=\dfrac{7250}{x}[/tex]
The force balance of the system is given by
[tex]ma=kx\\\Rightarrow m5g=\dfrac{7250}{x}\\\Rightarrow x=\dfrac{7250}{1070\times 5\times 9.81}\\\Rightarrow x=0.138\ \text{m}[/tex]
The distance the spring must be compressed is [tex]0.138\ \text{m}[/tex]
[tex]k=\dfrac{7250}{x^2}\\\Rightarrow k=\dfrac{7250}{0.138^2}\\\Rightarrow k=380697.33\ \text{N/m}[/tex]
The force constant of the spring is [tex]380697.33\ \text{N/m}[/tex].
