James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

874 Chapter 13 Vector Functions
The velocity of the projectile is
So its speed at impact is
vstd − r9std − 75s2 i 1 s75s2 2 9.8td j
| vs21.74d | − s(75s2 )2 1 (75s2 2 9.8 ? 21.74) 2 < 151 mys ■
Tangential and Normal Components of Acceleration
When we study the motion of a particle, it is often useful to resolve the acceleration into
two components, one in the direction of the tangent and the other in the direction of the
normal. If we write v − | v | for the speed of the particle, then
Tstd −
r9std
| r9std | − vstd
| vstd | − v v
and so
v − vT
If we differentiate both sides of this equation with respect to t, we get
5 a − v9 − v9T 1 vT9
If we use the expression for the curvature given by Equation 13.3.9, then we have
6 − | T9 |
| r9 |
− | T9 |
v
so
| T9 | − v
The unit normal vector was defined in the preceding section as N − T9y| T9 | , so (6) gives
and Equation 5 becomes
T9 − | T9 | N − vN
a T
FIGURE 7
T
a N
N
a
7 a − v9T 1 v 2 N
Writing a T and a N for the tangential and normal components of acceleration, we have
where
a − a T T 1 a N N
8 a T − v9 and a N − v 2
This resolution is illustrated in Figure 7.
Let’s look at what Formula 7 says. The first thing to notice is that the binormal vector
B is absent. No matter how an object moves through space, its acceleration always lies in
the plane of T and N (the osculating plane). (Recall that T gives the direction of motion
and N points in the direction the curve is turning.) Next we notice that the tangential
component of acceleration is v9, the rate of change of speed, and the normal component
of acceleration is v 2 , the curvature times the square of the speed. This makes sense if we
think of a passenger in a car—a sharp turn in a road means a large value of the curvature
, so the component of the acceleration perpendicular to the motion is large and the passenger
is thrown against a car door. High speed around the turn has the same effect; in
fact, if you double your speed, a N is increased by a factor of 4.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.