Determine the acceleration of P at time t.

Step 1: Determine the acceleration of P at time t. The force acting on particle P is given by F = [(3t^2 - 4t)i + (6t - 5)j] N. The particle has unit mass, so m = 1 kg. According to Newton's second law, F = ma. Since m=1, the acceleration a is equal to the force F. a(t) = (F)/(m) = ([(3t^2 - 4t)i + (6t - 5)j])/(1) a(t) = (3t^2 - 4t)i + (6t - 5)j m/s^2 The acceleration of P at time t seconds is (3t^2 - 4t)i + (6t - 5)j m/s^2. Step 2: Determine the velocity of P at time t seconds. Velocity is the integral of acceleration with respect to time. v(t) = a(t)\,dt = [(3t^2 - 4t)i + (6t - 5)j]\,dt v(t) = ( (3t^2 - 4t)\,dt)i + ( (6t - 5)\,dt)j v(t) = (t^3 - 2t^2 + C_1)i + (3t^2 - 5t + C_2)j We can write the constants of integration as a single vector C = C_1 i + C_2 j. v(t) = (t^3 - 2t^2)i + (3t^2 - 5t)j + C We are given that at t=3 seconds, the velocity is v(3) = (11i + 10j) m/s. Substitute t=3 into the velocity equation: v(3) = (3^3 - 2(3^2))i + (3(3^2) - 5(3))j + C v(3) = (27 - 18)i + (27 - 15)j + C v(3) = 9i + 12j + C Equating this to the given velocity: 11i + 10j = 9i + 12j + C C = (11i + 10j) - (9i + 12j) C = (11 - 9)i + (10 - 12)j C = 2i - 2j Substitute C back into the velocity equation: v(t) = (t^3 - 2t^2)i + (3t^2 - 5t)j + (2i - 2j) v(t) = (t^3 - 2t^2 + 2)i + (3t^2 - 5t - 2)j m/s The velocity of P at time t seconds is (t^3 - 2t^2 + 2)i + (3t^2 - 5t - 2)j m/s. Step 3: Determine the kinetic energy of P when it is moving parallel to the vector i. When the particle is moving parallel to the vector i, its j-component of velocity is zero. From the velocity equation in Step 2, set the j-component to zero: 3t^2 - 5t - 2 = 0 Factor the quadratic equation: (3t + 1)(t - 2) = 0 This gives two possible values for t: t = -(1)/(3) s or t = 2 s. Since time cannot be negative, we take t = 2 s. Now, find the velocity of P at t=2 s: v(2) = (2^3 - 2(2^2) + 2)i + (3(2^2) - 5(2) - 2)j v(2) = (8 - 8 + 2)i + (12 - 10 - 2)j v(2) = 2i + 0j So, v(2) = 2i m/s. The magnitude of the velocity is |v(2)| = sqrt(2^2 + 0^2) = 2 m/s. The kinetic energy KE is given by the formula KE = (1)/(2)mv^2. Given m=1 kg and v=2 m/s: KE = (1)/(2)(1 kg)(2 m/s)^2 KE =