$(i)$ Derive an expression for the kinetic energy of an object. Write the $SI$ unit of kinetic energy.
$(ii)$ An object of mass $10 \, kg$ is moving with a uniform velocity of $5 \, m s^{-1}$. Calculate the kinetic energy possessed by the object.

(D) $(i)$ Consider a body of mass $m$ initially at rest,i.e.,$u = 0$,on a frictionless surface. Let a constant force $F$ act on the body,producing an acceleration $a$. After a displacement $S$,the body attains a velocity $v$. Using the equation of motion $v^2 - u^2 = 2aS$,we get $a = v^2 / (2S)$.
The work done $W$ by the force is $W = F \times S$. Since $F = ma$,we have $W = (ma) \times S = m \times (v^2 / 2S) \times S = 1/2 mv^2$.
This work done is stored as kinetic energy $(KE)$. Thus,$KE = 1/2 mv^2$. The $SI$ unit of kinetic energy is Joule $(J)$.
$(ii)$ Given: Mass $m = 10 \, kg$,velocity $v = 5 \, m s^{-1}$.
$KE = 1/2 mv^2 = 1/2 \times 10 \times (5)^2 = 5 \times 25 = 125 \, J$.