Mathematics- mechanics

Question

Show that an object accelerating with acceleration a from velocity $u$ to velocity $v$, where $0<u<v$,over a time $t$ is traveling at a velocity of $(u+v)/2$ at time $t/2$; that is, that at the time halfway through the motion the velocity of the object is the mean of the initial and final velocities

Answer 1

Assuming constant acceleration, we can write it from the definition $$a=\frac{v-u}t$$ Calculating $w$, the velocity at $t/2$ $$w=u+a\frac t2=u+\frac{v-u}t\frac t2=\frac{u+v}2$$