Tom has built a large slingshot, but it is not working quite right. He thinks he can model the slingshot like an ideal spring with a spring constant of $k_0=65\mathrm{.}0\frac{N}{m}.$ However, when he pulls the slingshot back $0\mathrm{.}445m$ from a nonstretched position, it does not launch its payload as far as he wants.

His physics professor "helps" by telling him to aim for an elastic potential energy of $19\mathrm{.}5J.$ Tom decides he just needs elastic bands with a higher spring constant.

By what factor a does Tom need to increase the spring constant to hit his potential energy goal?

$a=$

During a follow$-$up conversation, Tom's physics professor suggests that he should leave the slingshot alone and try pulling the slingshot back further without changing the spring constant.

How many times $n$ further must Tom pull the slingshot back to hit the potential energy goal with the original spring constant?

$n=$

In which of the two scenarios does Tom have to pull harder?

increased pullback distance

they are equal

increased spring constant