the particle of mass m at rest is acted upon by force p for a time t. its kinetic energy after an interval t is

Explanation

Here's how we can find the kinetic energy of the particle:1. Work done by the force:The work done by the force (W) is given by the formula:W = F * dwhere:F is the magnitude of the force (given as P)d is the displacement of the particleHowever, in this case, the question specifies that the particle starts from rest. When an object starts from rest, its initial displacement is zero. Therefore, the work done by the force simplifies to:W = P * 0W = 02. Work-Energy principle:The work-energy principle states that the net work done on an object is equal to the change in its kinetic energy:W = ΔKEwhere:ΔKE is the change in kinetic energy (final kinetic energy - initial kinetic energy)Since the initial kinetic energy of the particle at rest is zero, the change in kinetic energy (ΔKE) becomes the final kinetic energy itself (KE).3. Finding the final kinetic energy:Combining the information from the above steps:0 = KE (because W = 0)This might seem like we cannot determine the kinetic energy. However, there's another approach.4. Using the relationship between force and acceleration:We know that the force acting on the particle (P) causes it to accelerate. The relationship between force, mass (m), and acceleration (a) is given by Newton's second law of motion:F = m * aRearranging this formula to solve for acceleration:a = F / m5. Relating acceleration and displacement:Although the question doesn't explicitly provide the displacement (d), we can use the relationship between acceleration, time (t), and final velocity (v_f) using the following kinematic equation:v_f = a * tSince the particle starts from rest, the initial velocity (v_i) is zero. Therefore, the final velocity (v_f) is the same as the change in velocity (Δv):Δv = v_f = a * t6. Relating kinetic energy and final velocity:The kinetic energy (KE) of the particle is related to its final velocity (v_f) by the formula:KE = 1/2 * m * v_f^27. Combining the equations:Substituting the expression for acceleration (a) from step 5 into the equation for final velocity (v_f) from step 6:v_f = (P / m) * tSquaring both sides to find the square of the final velocity (v_f^2):v_f^2 = (P^2 * t^2) / m^2Substituting this expression for v_f^2 into the formula for kinetic energy (KE) from step 6:KE = 1/2 * m * [(P^2 * t^2) / m^2]Simplifying the equation:KE = P^2 * t^2 / 2mTherefore, the kinetic energy of the particle after an interval 't' is P^2 t^2 / 2m. This matches answer choice b.