How can you derive an equation for the area under a velocity-time graph?
Kinematic equation of interest is
##v(t)=u+at## …..(1)
where ##v(t)## is velocity after time ##t##, ##u## is initial velocity of an object and ##a## is constant experienced by it.
Recall the expression
##”Displacement”=”Velocity”xx”time”##
Observe it looks like equation of a straight line in the form
##y=mx+c##.
We know that velocity is rate of change of displacement, therefore equation (1) can be written as
##(ds(t))/(dt)=u+at##
##=>ds(t)=(u+at)cdot dt## …..(2)
If we integrate both sides we get
##intds(t)=int_(t_0)^t (u+at)cdot dt##
##=>s(t)=int_(t_0)^t (u+at)cdot dt## ……(3)
We see that LHS of the equation is total displacement, and RHS is area under the velocity-time graph from time ##t_0## to ##t##.
Equation (3) is the required expression.
One should not be surprised if one calculates integral of RHS of equation (3) from time ##t=0## to ##t##, one actually obtains the other kinematic equation
##s=ut+1/2at^2##