12.3
Velocity and Acceleration
Projectile Motion
Examples
A baseball player at second base throws a ball 90
feet to the player at first base. The ball
is
thrown

at 50 miles per hour at an angle of 15 above the
horizontal. At what height does the player at first
base catch the ball if the ball is thrown from a
height of 5 feet?
A projectile is fired from ground level at an angle of 12

with the horizontal. Find the minimum initial
velocity if the projectile is to have a range of 150
feet.
12.4
Tangent Vectors and Normal
Vectors
Examples:
1) Find the unit tangent vector and find a set of
parametric equations for the line tangent to the
space curve at the point P.

r (t )  t , t ,
4 t2
P(1,1, 3 )
2) Find the principal unit normal vector to the
curve above at the specified point.
Sometimes shortcut to principal
unit normal vector.
For plane curves, you can find the principal unit
normal vector by finding T(t)= x (t )i  y (t ) j
and then observe that N(t) will either be
N2 (t )   y(t )i  x(t ) j OR
N2 (t )   y(t )i  x(t ) j
It will be the one that points toward the concave side
of the curve.
Examples:
1) Find T(t), N(t), aT and aN
for the given curve r(t)
2
r(t)= t i  2tj at t=1
t
t
t
e
cos
ti

e
sin
tj

e
k
r(t)=
at the given time
t=0