Answer
On the moon: $320$ sec, $66560$ ft.
On Earth: $52$ sec, $10816$ ft.
Work Step by Step
a. Step 1. Given $s(t)=832t-2.6t^2$ on the moon, we can find the velocity as $v(t)=s'(t)=832-5.2t$ m/s
Step 2. At the highest point $v(t)=0$ we have $832-5.2t=0$ thus $t=160$ sec
Step 3. Consider the symmetry, the time it takes to go up will be the same as it takes to fall down. Thus the total time the bullet will be aloft is twice that it takes to reach the maximum height, which means $T=2t=320$ sec.
Step 4. The maximum height it can reach happens at $t=160$; we have $s(160)=832(160)-2.6(160)^2=66560$ ft.
b. Repeat the above steps for the case on Earth:
Step 1. Given $s(t)=832t-16t^2$ on Earth, we can find the velocity as $v(t)=s'(t)=832-32t$ m/s
Step 2. At the highest point $v(t)=0$ we have $832-32t=0$; thus $t=26$ sec
Step 3. Consider the symmetry: the time it takes to go up will be the same as it takes to fall down. Thus the total time the bullet will be aloft is twice that it takes to reach the maximum height, which means $T=2t=52$ sec.
Step 4. The maximum height it can reach happens at $t=26$; we have $s(26)=832(26)-16(26)^2=10816$ ft.
