# A particle moves along a straight line and its position at time t is given by: 2t^3-18t^2+48t, t>=0?

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where s is measured in feet and t in seconds.
Use interval notation to indicate the time interval or union of time intervals when the particle is moving forward and backward. ...show more

Best Answer

velocity = derivative of position

v(t) = s'(t)

s(t) = 2t^3 - 18t^2 + 48t

s'(t) = 6t^2 - 36t + 48 = velocity

particle is moving forward when v(t) > 0 and backwards when v(t) < 0 find the zeros of the velocity function and then test points between the zeros (when particle has stopped to reverse direction) to find whether it is moving forwards or backwards.

v(t) = 6t^2 - 36t + 48 = 6[t^2 - 6t + 8] = 6(t-4)(t-2) v(t) = 0 at t = 4 and t = 2

point t = 1, (v1) = 6 - 36 + 48 > 0 so moving forwards

point t = 3, v(3) = 54 - 108 + 48 < 0 so moving backwards

point t = 5, v(5) = 150 - 180 + 48 > 0 so moving forwards

forward: between 0 and 2 and points greater than 4

backward: between 2 and 4

(I can't remember interval notation)

acceleration is the derivative of the velocity function

a(t) = v'(t)

v(t) = 6t^2 - 36t + 48

v'(t) = 12t - 36

a(t) = 12t - 36

find zero

12t-36 = 0, t = 3

plug in points before and after 3

point t = 0: 12(0)-36 <0 so slowing down

point t = 4: 48 - 36 > 0 so speeding up

Speeding up: points greater than t = 3

Slowing down: points between 0 and 3

v(t) = s'(t)

s(t) = 2t^3 - 18t^2 + 48t

s'(t) = 6t^2 - 36t + 48 = velocity

particle is moving forward when v(t) > 0 and backwards when v(t) < 0 find the zeros of the velocity function and then test points between the zeros (when particle has stopped to reverse direction) to find whether it is moving forwards or backwards.

v(t) = 6t^2 - 36t + 48 = 6[t^2 - 6t + 8] = 6(t-4)(t-2) v(t) = 0 at t = 4 and t = 2

point t = 1, (v1) = 6 - 36 + 48 > 0 so moving forwards

point t = 3, v(3) = 54 - 108 + 48 < 0 so moving backwards

point t = 5, v(5) = 150 - 180 + 48 > 0 so moving forwards

forward: between 0 and 2 and points greater than 4

backward: between 2 and 4

(I can't remember interval notation)

acceleration is the derivative of the velocity function

a(t) = v'(t)

v(t) = 6t^2 - 36t + 48

v'(t) = 12t - 36

a(t) = 12t - 36

find zero

12t-36 = 0, t = 3

plug in points before and after 3

point t = 0: 12(0)-36 <0 so slowing down

point t = 4: 48 - 36 > 0 so speeding up

Speeding up: points greater than t = 3

Slowing down: points between 0 and 3

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