Question

 A particle moving along the x axis . It’s position varies with time according to x(t)=-4t+2t^2

a) Find the average velocity between t=0s and t=2s

b) Find the average speed of the particle between t=0s and t=2s

Answer 1

a)

average velocity:

$v=\frac{x(2)-x(0)}{2-0}=\frac{0}{2}=0$

b)

distance travelled = length of curve x(t):

$L=\int^2_0\sqrt{1+(x'(t))^2}dt$

$x'(t)=4t-4$

$L=\int^2_0\sqrt{1+(4t-4)^2}dt=(\frac{arcsinh(4t-4)+(4t-4)\sqrt{16t^2-32t+17}}{8})|^2_0=$

$=2.32+2.32=4.64$

average speed:

$v=\frac{L}{\Delta t}=\frac{4.64}{2}=2.32$ m/s