Two particles on a line: direction and…

Name: ______________________ Date: ____________ Two particles on a line: direction and speed (9 points)

Table

Particles

P

and

Q

move along the

x

-axis for

0 \le t \le 6

. Their positions are given below, with

x

in meters and

t

in seconds.

Positions of particles P and Q at time t seconds, in meters
Particle	Position
$P$	$x_P(t)$ $= 2t^3 - 15t^2 + 24t + 5$
$Q$	$x_Q(t) = (t - 2)e^{-t/2}$

(a) Find the velocity of particle $P$ at time $t = 2$ . Is $P$ moving to the left or to the right at $t = 2$ ? Give a reason for your answer. [2 points]
(b) Is particle $P$ speeding up or slowing down at time $t = 2$ ? Give a reason for your answer. [2 points]
(c) Find all times $t$ , for $0 < t < 6$ , at which particle $P$ changes direction. Justify your answer. [2 points]
(d) Find the velocity $v_Q(t)$ of particle $Q$ . At time $t = 3$ , are particles $P$ and $Q$ moving in the same direction? Justify your answer. [3 points]

The question