# Projectile motion with Cauchy problem

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?
• It would be $\left\{\begin{matrix} x'(t)=V_0\cos\theta & \\ y'(t)=V_0\sin\theta -gt& \end{matrix}\right.$ wih initial conditions $x(t) = 0$ and $y(t) = h$ – Raducu Mihai May 27 at 14:45
• I think so. $x(t) = V_0\cos\theta t$ and $y(t) = V_0\sin\theta t + \frac{gt/2}{2}+h$ – Raducu Mihai May 27 at 22:12
• Not quite. You didn't integrate correctly and think about the signs. $x(t) = V_0 \cos (\theta) t$ and $y(t) = -V_0 \sin (\theta)t - \frac{1}{2}gt^2 + h$ – Tim Dikland May 29 at 9:30