Determine values using the information provided- displacement question

Question

Two objects are moving along a straight line.

The displacement in cm of each object at a time (t in seconds) is given by the equations below

Displacement of M :Sm(t)=at^2+bt+4

displacement of N: Sn(t)=t^3-9t^2+ct+d

Determine values of a,b,c and d using the following information

1. both objects have the same displacement when t=1
2. object N changes direction at t=2
3. both objects have the same initial acceleration at t=0
4. object M is at rest at t=3

Answer 1

S_m(t)=at²+bt+4

V_m(t)=2at+b

A_m(t)=2a

S_n(t)=t³-9t²+ct+d

V_n(t)=3t²-18t+c

A_n(t)=6t-18

When t=1, S_m= S_n

a+b+4=1-9+c+d --> a+b-c-d = -12

When t = 2, V_n = 0

12-36+c=0 --> c = 24

When t = 0, A_m=A_n

2a=-18 -->a=-9

When t=3, V_m=0

6a+b=0 --> b=-6a = -6(-9) = 54

a+b-c-d=-12

-9+54+24-d=-12

-d=-12+9-54-24=-81

d=81

a=-9

b=54

c=24

d=81