The second kinematic equation expresses displacement as a function of initial velocity, time, and constant acceleration. The equation is delta x equals v i times t plus one half a times t squared. This parabolic relationship shows how displacement increases quadratically with time when acceleration is present.
The second kinematic equation can be derived from basic principles. Starting with the definition of acceleration as the rate of change of velocity, and velocity as the rate of change of position. By integrating acceleration, we get velocity equals initial velocity plus acceleration times time. Substituting this into the velocity equation and integrating again gives us the displacement formula.
Let's solve a practical example. A car starts from rest and accelerates at 2 meters per second squared for 5 seconds. We need to find the displacement. Given initial velocity is zero, acceleration is 2, and time is 5 seconds. Using our kinematic equation, we substitute the values to get displacement equals 25 meters.
Now let's compare how different accelerations affect displacement. We'll examine three cases with accelerations of 1, 2, and 3 meters per second squared, all starting with an initial velocity of 1 meter per second. Notice how higher acceleration produces increasingly greater displacement over the same time period, creating steeper parabolic curves.
In summary, the second kinematic equation delta x equals v i t plus one half a t squared is fundamental to understanding motion with constant acceleration. It has wide applications in projectile motion, vehicle dynamics, free fall problems, and engineering design. The key insight is the quadratic relationship with time, where the linear term represents the contribution of initial velocity and the quadratic term represents the effect of acceleration.