Find the equation describing the motion of an object moving

Find the equation describing the motion of an object moving along a stright line when the acceleration is a = 3t, when the velocity at t=4s is 40 m/s, and when the object has traveled 86 m from orgin at t = 2s.

Solution

a=3t

hence dv/dt = 3t

=>dv = 3t dt

integrating both side weget:-

v= 3t²/2 + c

now at t=4 v=40

=>hence 40 = 3*4*4/2 + c

=>c = 40-24 = 16

now v=3t*t/2 + 16

now ds/dt = v =3t*t/2 + 16

now again integratin we get:-

t³/2 + 16t + d

now at t=2 s = 86

=>86 = 2*2*2/2 + 16*2 + d

=>d = 86-32-4 = 50

hence the equation of motion = t³/2 + 16t +50