Second Kinematic Equation

The second kinematic equation describes how an object's position changes over time when it experiences constant acceleration:

This equation tells us the position $x$ at any time $t$ by combining three parts:

• Initial position: $x_0$ (where the object starts)
• Motion from initial velocity: $v_i t$ (distance covered if velocity stayed constant)
• Effect of acceleration: $\frac{1}{2}at^2$ (extra distance from changing velocity)

Why the equation has this form:

Under constant acceleration, velocity changes linearly:

$v = v_i + at$

The position change equals the area under the velocity-time graph. This area forms a trapezoid that we can split into:

• A rectangle with area $v_i \cdot t$ (constant velocity contribution)
• A triangle with area $\frac{1}{2}(at) \cdot t = \frac{1}{2}at^2$ (acceleration contribution)

Adding these areas to the initial position $x_0$ gives us the complete equation:

$$x = x_0 + v_i t + \frac{1}{2}at^2$$

Play the visualization below to see how the area under the velocity graph contributes to this equation.