How do you find the distance travelled?

Question

The velocity function (in meters per second) for a particle moving along a line is given by v(t)=(t^3)-(4t^2) Find the displacement and the distance traveled by the particle during the time interval [-1,5]. Your answers require that you enter the correct units.

Answer 1

How do you find the distance travelled? The velocity function (in meters per second) for a particle moving along a line is given by v(t) = (t³) – (4t²) Find the displacement and the distance traveled by the particle during the time interval [-1,5]. Your answers require that you enter the correct units.

Detailed Solution:

Known Facts:

* Derivative of distance s(t) is velocity v(t): ds(t)/dt = v(t)

* Derivative of velocity v(t) is acceleration a(t): dv(t)/dt = a(t)

* Integration of velocity v(t) is distance s(t): ∫ v(t) dt = s(t)

* Integration of acceleration a(t) is velocity v(t): ∫ a(t) dt = v(t)

Given: v(t) = (t³) – (4t²) find s(t) in time interval [a, b] = [-1, 5]<=Integration of velocity v(t) is distance s(t):

s(t) = ∫_a^b v(t) dt ==> s(t) = ∫_-1⁵ (t³) – (4t²) dt ==> s(t) = (t⁴/4) – (4t³/3)|_-1⁵ ==>

s(t) = [((5)⁴/4) – (4(5)³/3)] – [((-1)⁴/4) – (4(-1)³/3)] = [(625/4) – (500/3)] – [(1/4) – (-4/3)]

= (625/4) – (500/3) – (1/4) – (4/3) = (624/4) – (504/3) = 156 – 168 = -12 meters