Final answer:
To solve this problem, we can use the principle of conservation of mechanical energy. Initially, the block is compressed against the first spring and gains potential energy. When it reaches the second spring, this potential energy is converted into compression potential energy of the second spring. Using the conservation of mechanical energy, we can calculate the compression distance (d2) of the second spring which is equal to 0.080 m.
Step-by-step explanation:
To solve this problem, we can use the principle of conservation of mechanical energy. Initially, the block of mass m is compressed against the first spring. When it is released, it gains kinetic energy and potential energy due to the compression of the second spring. We can set up an equation to solve for the compression distance d2 of the second spring.
First, we calculate the potential energy of the block-spring system when it is just released from the first spring:
PEinitial = (1/2)k1x2 = (1/2)(549 N/m)(0.1 m)2 = 2.745 J
The kinetic energy of the block-spring system when it reaches the second spring is given by:
KE = (1/2)mv2
To find v, we can use the conservation of mechanical energy:
PEinitial = KE
(1/2)k1x2 = (1/2)mv2
Simplifying the equation, we get:
v = sqrt((k1x2)/m)
Now that we have the velocity of the block, we can calculate the potential energy of the block-spring system when it compresses the second spring:
PEfinal = (1/2)k2d22
Using the conservation of mechanical energy, we can equate the initial and final potential energies:
(1/2)k1x2 = (1/2)k2d22
Simplifying the equation, we get:
d2 = sqrt((k1x2)/k2)
Plugging in the values:
d2 = sqrt(((549 N/m)(0.1 m)2)/(295 N/m)) = 0.080 m
Therefore, the second spring will compress by a distance of 0.080 meters when the block runs into it.