Question

Question #089b1

Answer 1

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 acceleration experienced by it.

1. Recall the expression
   `"Displacement"="Velocity"xx"time"`
2. 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`