prove that change in kinetic energy of a particle is equal to the work done on it by a variable force

The work-energy theorem states that the work done on a body is equal to the change in kinetic energy of the body.

Suppose a body of mass ‘m’ is under an acceleration ‘a’ due to which its initial speed ‘u’ at time t = 0 becomes final velocity ‘v’ at time t = t.

The distance travelled by the body in this time can be found out using,

v² = u² + 2as

Multiplying both side by ½ m we have,

½ mv² = ½ mu² + mas

= > (ma)s = ½ mv² – ½ mu²

[Force on the body = ma]

= > Work done = change in kinetic energy of the body