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

Question

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

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

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

(d) Identify the time(s) at which the particle changes direction. (Enter your answers as a comma-separated list.)

Answer 1

a) v = ds/dt so, taking the derivative v(t) = 3t² - 10t + 3

b) if the velocity is moving in the positive direction then v(t) is positive. To find the places where velocity MIGHT change directions, set v(t) equal to zero (zero velocity means the object is standing still)

0 = 3t² - 10t + 3

0 = (3t -1)(t - 3)

3t - 1 = 0 when t = 1/3 and t - 3 = 0 when t = 3

Setting up a number line, we can try times in each of the three intervals to see what the velocity is doing in each one. For v < 1/3 I tried t = 0 and got a positive value for velocity (3•0² - 10•0 + 3 = 3), for v between 1/3 and 3 I tried t = 1 and got a negative result (3•1² - 10•1 + 3 = -4), and for t>3 I tried t = 4 and got a positive result (3•4² - 10•4 + 3 = 11) and put those results on the number line:

So the particle is moving in the positive direction for [0,1/3) and from (3,∞)

The particle is moving in the negative direction for (1/3,3)

The particle changes direction at t = 1/3 and t = 3 or t = 1/3, 3