(ii)Use the average velocity to derive S= ut + 1/2 at2 (4 marks)

For an object moving with constant acceleration, the average velocity ($\bar{v}$) is the arithmetic mean of its initial velocity ($u$) and final velocity ($v$).

$$\bar{v} = \frac{u+v}{2}$$

Displacement ($S$) is the product of average velocity and time ($t$).

$$S = \bar{v}t$$

Substitute the expression for $\bar{v}$ from Step 1 into this equation:

$$S = \left(\frac{u+v}{2}\right)t$$

The first equation of motion states that the final velocity ($v$) is equal to the initial velocity ($u$) plus the product of acceleration ($a$) and time ($t$).

$$v = u + at$$

Substitute $v = u + at$ into the equation for $S$ from Step 2:

$$S = \left(\frac{u + (u+at)}{2}\right)t$$

$$S = \left(\frac{2u+at}{2}\right)t$$ $$S = \left(\frac{2u}{2} + \frac{at}{2}\right)t$$ $$S = \left(u + \frac{1}{2}at\right)t$$

Distribute $t$:

$$S = ut + \frac{1}{2}at^2$$

This is the required kinematic equation.