A particle of mass m at rest is acted upon by a force P for time t. Its kinetic energy is:

Explanation

The kinetic energy of a particle is given by the formula: KE = ½ mv² In this case, the particle is initially at rest, so its initial velocity (vo) is zero. The force (P) acting on the particle for time t will cause it to accelerate. According to Newton's second law, the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). Using this relationship, we can express the acceleration (a) in terms of force (P) and mass (m): P = ma Solving for acceleration: a = P/m Using the kinematic equation: v = vo + at Since the initial velocity is zero: v = at Now, we can substitute the expression for acceleration into the equation for velocity. v = (P/m)t Finally, we can calculate the kinetic energy by using the formula: KE = ½ mv² Putting values in the equation, we get: KE = ½ m ((P/m)t)² = ½ mt² (P²/m²) KE = (P²/t²) / 2m