If the displacement of a particle is given by the formula S = 3t^3 - 20t^2 + 40t. Find (i) the displacement after 3 seconds (ii) the formula for the velocity at any time t (iii) the value of t when the particle is not moving (iv) the initial velocity of the particle (v) the formula for the acceleration at any time t (vi) the initial acceleration of the particle.​

At t = 0   S = 0

S = 3 * 3^3 - 20 * 3^2 + 40 * 3    after 3 sec

S = 3 * 27 - 20 * 9 + 120 = 21 after 3 sec

v = ds/dt = 9 * t^2 - 40 * t + 40 = 9 * 9 - 40 * 3  + 40 = 1

ds/dt = 9 t^2 - 40 t + 40      v at any time

If zero = 9 t^2 - 40 t + 40  then you need to solve the quadratic for t

a = ds^2 / dt^2 = dv /dt = 18 t - 40    for the acceleration at any time

At t = zero (the initial acceleration) a = -40

force is mass*acceleration.

we have a very small mass: 11.4*0.001 kg

if we want 0.035n of force, we need an acceleration:

0.035 = 11.4*0.001*a

a = 0.035/(11.4*0.001) m/s^2