A particle's acceleration is given by a=(10-t) m/s². It starts from rest. If its velocity is 2 m/s when t=1 sec, find its velocity and displacement in terms of t.

You're on a roll — Given: Acceleration a = (10-t) m/s^2. A particle starts from rest (implies s=0 at t=0). Velocity V = 2 m/s when t = 1 sec. a) (i) Express its velocity in terms of t. Step 1: Integrate acceleration to find velocity. Acceleration is the rate of change of velocity, so a = (dv)/(dt). v = a \, dt v = (10-t) \, dt v = 10t - (t^2)/(2) + C_1 Step 2: Use the given condition to find the constant of integration C_1. We are given that v = 2 m/s when t = 1 sec. 2 = 10(1) - (1^2)/(2) + C_1 2 = 10 - 0.5 + C_1 2 = 9.5 + C_1 C_1 = 2 - 9.5 C_1 = -7.5 Step 3: Write the velocity equation. v(t) = 10t - (t^2)/(2) - 7.5 The velocity in terms of t is v(t) = 10t - (t^2)/(2) - 7.5 m/s. a) (ii) Its displacement after t seconds. Step 1: Integrate velocity to find displacement. Velocity is the rate of change of displacement, so v = (ds)/(dt). s = v \, dt s = (10t - (t^2)/(2) - 7.5) \, dt s = 10(t^2)/(2) - (1)/(2)(t^3)/(3) - 7.5t + C_2 s(t) = 5t^2 - (t^3)/(6) - 7.5t + C_2 Step 2: Use the initial condition to find the constant of integration C_2. The particle "starts from rest", which implies that its initial displacement is 0 at t=0. 0 = 5(0)^2 - (0^3)/(6) - 7.5(0) + C_2 C_2 = 0 Step