It takes energy to compress a spring. This energy is stored as spring potential
energy, which can be calculated using: E_{spring} = 1/2 K x^{2}, where K
is the spring constant
and x is the distance
the spring is compressed. 
At a dock, a boat with a mass of 50,000 kg hits a bumper supported by two springs that stop the boat and absorb its kinetic energy. 
Determine the spring constant of the springs that is required if the maximum compression is to be 60 cm for a boat speed of 2.4 m/s. 


Read: 
It is important to
note that two springs are used to stop the vehicle.
All of the initial kinetic energy of the vehicle must be absorbed by the
springs and converted to spring potential energy. 










Given: 

Eqn 1 

K =
spring constant 








x =
displacement of the spring, in this case compression. 











m 
50000 
kg 







v 
2.4 
m/s 



g_{c} 
1 
kgm/Ns^{2} 

x 
0.6 
m 
















Find: 
K 
??? 
N/m 
















Assumptions: 
1 The spring is a linear
spring and therefore the given
equation applies. 




2 All of the kinetic
energy of the boat is absorbed by the springs. 












Equations
/ Data / Solve: 


















The key to solving
this problem is to recognize that the final potential energy of the two
springs must be equal to the initial kinetic energy of the vehicle. So, we should begin by calculating the
initial kinetic energy of the boat. 


















Eqn 2 










Because there are two
identical springs : 



Eqn 3 

Plugging given values
into Eqn 2 and Eqn 3 yields: 

E_{kin} 
144000 
J 







E_{spring} 
72000 
J 











Next, we can solve Eqn 1 algebraically for the spring
constant, K. 

Eqn 4 











First, let's work on the
units. 



Eqn 5 











Now, let's calculate the
value of the spring constant. 

K 
4.00E+05 
N/m 










Answers: 
K 
4.00E+05 
N/m 















