Analytical Solutions


Suppose there is a ball moving with velocity $(15, 15)$ at the origin $(0, 0)$. If the acceleration due to gravity is (0, -9.8), what will the trajectory look like?

原点$(0, 0)$に速度$(15, 15)$で移動するボールがあるとします。重力による加速度を$(0, -9.8)$とすると、このボールはどんな軌跡を描くでしょうか。

If $v_0$ is the initial velocity, $a$ is the acceleration, the velocity $v_t$ at the time $t$ is


$v_t = v_0 + at$

Integrating😇 this, the position $p_t$ of the ball at the time $t$ is:


$p_t = v_0t + \frac12at^2$

Solving a problem in this way so that the values can be calculated at once is called "solving analytically”. In this case, the position of the ball at any time (even in the past) can be obtained with a single calculation as long as $t$ is determined.


Discrete time