Question

The displacement, x, of a piston in a vehicle with an experimental engine accelerating uniformly, with respect to a stationary point on the ground, is modelled using the function

 x(t)=0.5t+t²-8sin5t  

where t is time, and shown in Figure 5 below.

       Figure 5 Displacement of a piston in a moving car as a function of time

a.     Differentiate the function above and write an expression for the velocity, v, of the piston as a function of time t. 

 

b.     Apply the Newton–Raphson method to x(t) to find the point between t=1s and t=1.5s where x(t) is zero to an accuracy better than 0.001. Give your answer to three decimal places. 

(Hint: there are a number of points where x(t) is zero, as can be seen from the graph. The Newton-Raphson method can converge to any of those depending on the initial value. Try t=1.5s as your starting guess.) What value of t does the method converge to if your initial value is 1.0 s? 

c.     Another way to find t when x=0 is to write

0=0.5t+t²-8sin5t

and from this we can write

 0.5t+t²=8sin5t     

Plot y=0.5t+t² and y=8sin(5t) on the same graph paper, between t=1.0s and t=1.5s (or use a spreadsheet plotting software) and estimate t from where the two lines cross.