5128_Ch03_pp098-184

132 Chapter 3 Derivatives
–1
6
4
2
–2
–4
s(t)
2 4 6
Figure 3.27 The graphs of
s(t) t 2 4t 3, t 0 (blue) and its
derivative v(t) 2t 4, t 0 (red).
(Example 5)
X 1T =T2-4T+3 Y 1T =2
t
SOLUTION
(a) The displacement is given by s(2) s(0) (1) 3 4. This value means that
the particle is 4 units left of where it started.
(b) The average velocity we seek is
s(4 ) s(0)
3
3
0 m/sec.
4 0 4
(c) The velocity v(t) at any time t is v(t) ds/dt 2t 4. So v(4) 4 m/sec
(d) The acceleration a(t) at any time t is a(t) dv/dt 2 m /sec 2 . So a(4) 2.
(e) The graphs of s(t) t 2 4t 3 for t 0 and its derivative v(t) 2t 4 shown in
Figure 3.27 will help us analyze the motion.
For 0 t 2, v(t) 0, so the particle is moving to the left. Notice that s(t) is decreasing.
The particle starts (t 0) at s 3 and moves left, arriving at the origin
t 1 when s 0. The particle continues moving to the left until it reaches the point
s 1 at t 2.
At t 2, v 0, so the particle is at rest.
For t 2, v(t) 0, so the particle is moving to the right. Notice that s(t) is increasing.
In this interval, the particle starts at s 1, moving to the right through the origin
and continuing to the right for the rest of time.
The particle changes direction at t 2 when v 0.
(f) Enter X 1T T 2 4T3, Y 1T 2 in parametric mode and graph in the window
[5, 5 ] by [2, 4 ] with Tmin 0, Tmax 10 (it really should be ), and
XsclYscl1. (Figure 3.28) By using TRACE you can follow the path of the particle.
You will learn more ways to visualize motion in Explorations 2 and 3.
Now try Exercise 19.
T=0
X=3
Y=2
[5, 5] by [2, 4]
Figure 3.28 The graph of X 1T
T 2 4T 3, Y 1T 2 in parametric
mode. (Example 5)
EXPLORATION 2
Modeling Horizontal Motion
The position (x-coordinate) of a particle moving on the horizontal line y 2 is
given by xt 4t 3 16t 2 15t for t 0.
1. Graph the parametric equations x 1 t 4t 3 16t 2 15t, y 1 t 2 in 4, 6
by 3, 5. Use TRACE to support that the particle starts at the point 0, 2,
moves to the right, then to the left, and finally to the right. At what times does
the particle reverse direction?
2. Graph the parametric equations x 2 t x 1 t, y 2 t t in the same viewing
window. Explain how this graph shows the back and forth motion of the particle.
Use this graph to find when the particle reverses direction.
3. Graph the parametric equations x 3 t t, y 3 t x 1 t in the same viewing
window. Explain how this graph shows the back and forth motion of the particle.
Use this graph to find when the particle reverses direction.
4. Use the methods in parts 1, 2, and 3 to represent and describe the velocity of the
particle.