A particle moves along a straight line with velocity given (in metres per second) by
v(t)=1+t3t2,0≤t≤2.
a) Write a definite integral that represents the exact distance the particle travels during the time interval 0≤t≤2.
b) Choose a new variable u(t) that renders the integrand a rational function of u. Provide a clear justification for your choice, citing the relationship between du and the factors in v(t)dt.
c) Perform the variable change, convert the limits, and evaluate the integral found in part (a) to obtain the exact distance travelled.
d) Interpret the numerical answer in physical terms, including appropriate units.