An archer shoots a $0\mathrm{.}024-$kg arrow at a target with a speed of $54$ m$/$s$.$ When it hits the target, it penetrates to a depth of $0\mathrm{.}083$ m$.($a$)$ What was the average force exerted by the target on the arrow? $($b$)$ If the mass of the arrow is doubled, and the force exerted by the target on the arrow remains the same, by what multiplicative factor does the penetration depth change? Explain.