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Question

The function s(t) describes the motion of a particle along a line.

s(t) = 8t − t²

(a) Find the velocity function v(t) of the particle at any time t ≥ 0.

v(t)=

(b) Identify the time interval on which the particle is moving in a positive direction. (Enter your answer using interval notation.)

(c) Identify the time interval on which the particle is moving in a negative direction. (Enter your answer using interval notation.)

(d) Identify the time at which the particle changes direction.

t=

Answer 1

s(t) = 8t − t²

(a) Find the velocity function v(t)  of the particle at any time t ≥ 0.

v(t) = d/dt (s(t)) = d/dt (8 t - t²)

= 8 - 2t

 

 

(b) Identify the time interval on which the particle is moving in a positive direction. (Enter your answer using interval notation.)

 for 0< t <4, the particle moves in the positive direction, since in this interval the v(t) is positive.

 

(c) Identify the time interval on which the particle is moving in a negative direction. (Enter your answer using interval notation.)

for t > 4 the particle is moving in the negative direction.

 

 

(d) Identify the time at which the particle changes direction.

t = 4

Answer 2

s(t) = 8t-t^2

v(t) = s'(t) = 8 -2t

8-2t>0

8>2t

4>t

t < 4 seconds.

it's moving in a positive direction when 0<t<4 or [0,4)

it's moving in a negative direction when t>4 (4, infinity)

at t= 4 seconds, the particle changes direction