Displacement when acceleration is a function of time
We will consider a particle moving along a straight line with speed given by
v(t) = v0 +a(t)t (1)
With
a(t) = a0t (2)
which means that acceleration is a linear function of time – increasing linearly with time.
a0 is the rate of change of acceleration. Its dimension
[a0] = [a(t)] / [t] = m/s2/s = m/s3
The speed at time t from the beginning of this motion is
v(t) = v0 +a0tt = v0 + a0t2 (3)
The distance traveled during time t can be found from the formula
(4)
Solving these two integrals we get
D = v0t + (1/3) a0t3 (5)
If you are not familiar with calculus do not bother yourself with formula (4), simply use, if necessary, formula (5) to calculate displacement if acceleration is a linear function of time.
The graphical interpretation of formula (5) is on the figure below. The distance traveled is equal to the ruled area on the picture.