Make a strongarm stronger

As you can see, the two springs are connected parallel.

If two springs are connected parallel, then the formula for their stiffnesses is:

$\boxed{k = k_{1} + k_{2}}$

$k = k + \frac{1}{2}k$

$\boxed{k = \frac{3}{2}k}$

The reason why we can add the springs strength together, can be easily explained when we consider what the strength means. The force the spring employs when compressed or stretched a certain distance from its equilibrium is

$f = k \cdot x$

Therefore, the force employed by the two springs together is

$f = k \cdot x + \frac {k}{2} \cdot x$

Which then is equivalent to

$f = \frac {3k}{2} \cdot x$

And now we see that we can consider the two springs, when employed in parallel, as one spring with the strength the sum of their strengths.

The forces due to both the springs would add up for any displacement. k1+k2=K

This problem is really easy to overthink, but it is actually quite simple.
k = 1 k
1/2 k = .5 k
Now if we add these two “conversions” together you get 1.5 k !
But one more switch-a-roo is necessary to complete the job.
1.5 k = 3/2 k
Therefore the answer is:
3/2 k !

The words " the ease with which it can be modified to shoot faster and farther. " give the solution without any calculus. The only solution that increase significantly the strength of the shoot is the first one ;-) Do you really think that people will notice a 1.22 difference ? :-P

So in order to get the total strength of the springs, we add the individual strengths together. See below :

k + 1/2 k

= k/1+ k/2 (LCM is 2)

= (2k+k)/2

= 3k/2 or 3/2 k

$k=1k$ , so $\frac{1}{2}$ $k$ $+1k=$ $1$ $\frac{1}{2}$ $k$ which would be * $\frac{3}{2}$ $k$ *