I have the following problem:
An airplane P, flying at a height of $h$ needs to hit the target on the ground $T$. The airplane is flying horizontaly at a constant speed $V_0$.
Find the Cauchy problem that $x()$ and $y()$ must satisfy, knowing that only gravity acts on the projectile.
At what distance $x^*$ must the projectile be launched in order to hit the target? After what amount of time $t^*$ will the projectile hit the target?
I know a little about projectile motion and I am very confused at the part where the Cauchy problem is asked? What is that asking? Is it asking to separate the horizontal and vertical movement in two function $x$ and $y$ ? I don't want you to solve my problem for me but any hint would be very muh appreciated. Thank you!
The way I interpret the problem it the following. Consider the regular x-y plane. At time $t=0$ we have a projectile at $x=0$ and $y=h$. Since it also has the same velocity of the plane it will move in the positive $x$-direction with velocity $V_0$ when $t \geq 0$. At $t=0$ we also know that the projectile is at $y=h$ and has zero velocity. However it will get some velocity in the negative $y$-direction when the time starts running, since gravity is acting on it.
Now it is you task to find the $x$ and $y$ coordinate of the projectile for $t > 0$. Thus filling in $$ \begin{aligned} y(t) &= ? \\ x(t) &= ?\end{aligned}$$ Taking into account the things we know from the situation at $t=0$. Can you do this?