Question

A particle moves along a line. The particles position,s, in centimetres at t seconds is modelled by s(t)=t^3-9t^2+24t+20,where t is greater than or equal to zero. What is the total distance travelled by the particle in the first 8 seconds??

Answer 1

We know that the expression \[s(0) +\int_{0}^{8} |v(t)| dt\] will give the total distance traveled.

We can see from the position function that \[s(0)=20\].

From the graph of the velocity function, we can evaluate the definite integral by breaking it into parts "above" and "below" the x-axis. The graph of velocity crosses the x-axis at x=2 and x=4. Thus, we can find the total distance traveled by evaluating the following expression:

\[20+\int_{0}^{2} v(t) dt - \int_{2}^{4} v(t) dt + \int_{4}^{8} v(t) dt\]