Choose a polynomial a(t) of any order any with arbitrary coefficients to represe
ID: 2132101 • Letter: C
Question
Choose a polynomial a(t) of any order any with arbitrary coefficients to represent the instantaneous acceleration of an object moving along a straight line. Also pick up its initial position (in meters) and initial velocity (in meters/seconds).
(a) Determine the analytical expression for the instantaneous velocity of the object;
(b) Determine the analytical expression for the position of the object;
(c) Select any time instant and describe the motion of the object at that instant specifying whether it moves in the positive/negative direction, or is at rest, also whether it speeds up/slows down or moves with constant velocity.
Explanation / Answer
a) suppose a(t) = a1.t^x
and initial velocity v(o) = v0
initial position r(0) = r0
as we know dv/dt = a(t) = a1.t^x
dv = a(t).dt = a1.t^x .dt
integrating we get ,
v(t) = a1.t^(x-1) / (x-1) + a2
V(0) = a2 =v0
s0 , v(t) = a1.t^(x-1) / (x-1) + v(0)
b)
as we know dr/dt = v(t) = a1.t^(x-1) / (x-1) + v(0)
dr = v(t).dt = [a1.t^(x-1) / (x-1) + v(0) ].dt
integrating we get ,
r(t) = a1.t^(x-2) / (x-2)(x-1) + v0.t + a3
r(0) = a3 =r0
s0 ,r(t) = a1.t^(x-2) / (x-2)(x-1) + v0.t + r0
c)
calculate v(t) : if zer0 then at rest
+ve ...moving in+ve direction
-ve ...moving in -vs direction
diffrenciate the v(t)
dv(t) /dt
if dv(t) /dt is zer0. .....speed is constant.
dv(t)/dt is +ve ....speeding up
dv(t)/dt is -ve .....speeding down