Names of all group members:

Recitation section:

Pledge and Signatures:

Required The science base on a small moon has detected an incoming asteroid. The only things standing

between you and fiery death is your calculus skill.

The gravitational constant on the moon is $g=2.$ There is a defense cannon at $(0,0)$ that launches

projectiles at a fixed speed $||vec(v)||=20k\frac{m}{s}\mathrm{.}100km$ away, at $(100,0),$ is the science base. The science

base has established that the asteroid has the following postion function.

$x_a(t)=-4t-100$

$y_a(t)=-8t-400$

The asteroid will hit at $t=-50,$ i$.$e$.50$ seconds before midnight tonight it will pass through $(100,0).$

You must find a firing solution, i$.$e$.$ an angle $$ and a time $t_{\text{f}\text{i}\text{r}\text{e}}$ at which to fire the projectile. The

Defense Cannon Manual suggests the following step by step approach.

$($a$)$ Pick an angle $$ and assume the projectile is launched at $t=0.$ Determine whether the curves

of the asteroid and the projectile cross. Give the times $t_a$ when the asteroid reaches that point

and the time $t_p$ when the projectile reaches that point. $($The numbers will get messy, it's okay

to use a calculator and decimal numbers.$)$

REMINDER. If the projectile is fired from $(0,0)$ at $t=0$ with initial velocity $($:$v_1,v_2$:$),$ then at the

time $t$ the position of the projectile is given by $x(t)=v_{1}t,y(t)=v_{2}t-\frac{g{t}^2}{2}.($You may

remember this from a Mechanics course, and we will derive this in a lecture this week, but for

now let's take it on faith.$)$

$($b$)$ From what you found in part $($a$),$ determine the time $t_{\text{f}\text{i}\text{r}\text{e}}$ at which the projectile should be

fired to hit the asteroid at that point. Provide explanation $($the cannon's AI will refuse solution

with incomplete work.$)$