There are two springs, one long spring of constant k1 and it is inside a smaller spring of constant k2. Both springs follow Hooke's Law. A box of mass M moves without friction and compresses spring 1 a distance d1 where it hits spring 2. Spring 1 and 2 then compress further distance d2. Solve for the initial velocity of the block.
F = -kx
U = 1/2 mv^2 = 1/2 kx^2
The Attempt at a Solution
I was able to get the solution from the following: (1/2)M v^2 = 1/2 k1 d2^2 + 1/2 k2 (d2-d1)^2 and solving for v. I also thought I could get the solution from finding the area under a force vs distance curve but I was unsuccessful. In this attempt I found the area under the small triangle formed by k1 slope, the area of the larger triangle formed by k1+k2 and the little rectangle beneath.
The equation was: 1/2 k1 d1^2 + 1/2 [(k1+k2)d2 - k1d1]*(d2-d1) + k1d1d2
I attached a picture of the initial scenario and graph. Any help is greatly appreciated, thanks!