Why Physics formulas
Kinematic Equation for Displacement
\(d = v_0t + \frac{1}{2}at^2\)
d is distant that object has traveled after t; v0 is the velocity when object starts moving; a is the constant acceleration
1. Calculus
Acceleration a is the derivative of velocity v with respect to time t:
\(a = \frac{dv}{dt}
\)
\(dv = a.dt\)
Integrate both sides with respect to time
\( \int dv\ = \int_{0}^{t} a \,dt\)
\(v = at + v_0\)
Velocity is the derivative of displacement d with respect to time:
\(v = \frac{dd}{dt}\)
\(dd = v.dt = (v_0+at)dt\)
\(\int dd = \int_{0}^{t}(v_0+at)\,dt\)
\(d = v_0t+\frac{1}{2}at^2\)