A particle moves with acceleration a = (4-t) ext m/s2. Its initial velocity is v0 = 3 ext m/s. Find its velocity, displacement, and time to attain maximum velocity.

Step 1: Understand the given information. The acceleration of the particle is given by a = (4-t) m/s^2. The initial velocity (at t=0) is v_0 = 3 m/s. a) (i) Express the velocity v in terms of t. Velocity is the integral of acceleration with respect to time: v = a\,dt. v = (4-t)\,dt v = 4t - (t^2)/(2) + C To find the constant of integration C, use the initial condition: when t=0, v=3. 3 = 4(0) - ((0)^2)/(2) + C 3 = C Substitute C=3 back into the velocity equation: v = 4t - (t^2)/(2) + 3 The velocity v in terms of t is v = 4t - (t^2)/(2) + 3 m/s. a) (ii) Find the velocity of the particle after 3 seconds. Substitute t=3 into the velocity equation: v = 4(3) - ((3)^2)/(2) + 3 v = 12 - (9)/(2) + 3 v = 15 - 4.5 v = 10.5 m/s The velocity after 3 seconds is 10.5 m/s. b) (i) Calculate the time taken to attain maximum velocity. Maximum velocity occurs when the acceleration is zero (a=0). a = 4-t Set a=0: 0 = 4-t t = 4 s The time taken to attain maximum velocity is 4 s. b) (ii) Calculate the displacement of the particle by the time it attains the maximum velocity. Displacement s is the integral of velocity with respect to time: s = v\,dt. We use the velocity equation v = 4t - (t^2)/(2) + 3. s = (4t - (t^2)/(2) + 3)\,dt s = (4t^2)/(2) - (t^3)/(2 · 3) + 3t + D s = 2t^2 - (t^3)/(6) + 3t + D Assume initial displacement is zero: when t=0, s=0. 0 = 2(0)^2 - ((0)^3)/(6) + 3(0) + D 0 = D So, the displacement equation is: s = 2t^2 - (t^3)/(6) + 3t The particle attains maximum velocity at t=4 s. Substitute t=4 into the displacement equation: s = 2(4)^2 - ((4)^3)/(6) + 3(4) s = 2(16) - (64)/(6) + 12 s = 32 - (32)/(3) + 12 s = 44 - (32)/(3) To combine these, find a common denominator: s = (44 × 3)/(3) - (32)/(3) s = (132 - 32)/(3) s = (100)/(3) m The displacement of the particle by the time it attains maximum velocity is (100)/(3) m or 33(1)/(3) m. That's 2 down. 3 left today — send the next one.