What kind of formula should I use to get the impact time of two moving objects?

Question

So I have the initial and end positions of two objects that move in different directions and I want to know where will they collide. How can I know if two complex shapes will collide? I don´t know much about physics and am having a hard time trying to understand it. If these shapes move in their respective directions without a change in speed, where will they collide? I just don´t get how to be aware of their physical forms when they collide.

Answer 1

Usually, we assume that the bodies are just a moving point in the space. So you don't need to worry about their shape; they will collide if there is a time $t_{col}$ in which they are both at the same point of space.

In order to know if that will happen, you need to write the equation of movement of each of the two bodies, $x_1=x_1(t)$ and $x_2=x_2(t)$. The specific form of the equation depends on what kind of movement we consider (linear motion, circular motion, etc...).

In this case, we are dealing with a linear motion with constant velocity. In order to compute the velocity of each body you need to use the formula $$ \vec{v} = \dfrac{\vec{x}(t_f)-\vec{x}(t_i)}{t_f-t_i}, $$ where $t_i$ is the initial time (usually $0$) and $t_f$ is the final time, and $\vec{x}(t_i),\vec{x}(t_f)$ are the positions at such times.

Now, the equation of movement of linear motion with constant velocity is \begin{equation}\label{linear} \vec{x}(t) = \vec{x}(t_i)+\vec{v}\cdot t. \end{equation} Write this equation for both bodies, and try solving $x_1(t)=x_2(t)$ for $t$. If there's no solution to the equation, that means that the bodies won't collide. If there's a solution $t_{col}$, then the collision will take place at $x_1(t_{col})$ (or equivalently at $x_2(t_{col})$).