MATH 2203 –Exam 2 (Version 2) Solutions
September 28, 2005
S. F. Ellermeyer Name
Instructions. Your work on this exam will be graded according to two criteria: mathe-
matical correctness and clarity of presentation. In other words, you must know what
you are doing (mathematically) and you must also express yourself clearly. In particular,
write answers to questions using correct notation and using complete sentences where
appropriate. Also, you must supply su¢ cient detail in your solutions (relevant calculations,
written explanations of why you are doing these calculations, etc.). It is not su¢ cient to just
write down an “answer” with no explanation of how you arrived at that answer. As a rule
of thumb, the harder that I have to work to interpret what you are trying to say, the less
credit you will get. You may use your calculator but you may not use any books or notes.
1. If two objects travel through space along two di¤erent curves, it’s often important to
know whether the two objects will collide. The two paths of motion might intersect,
but we need to know whether the two objects are in the same place at the same time.
Suppose that the trajectories of two di¤erent particles are given, respectively, by the
vector–valued functions
r1(t) = D3;p3t; 6t2+8p312t+ 14 8p3E
and
r2(t) = D3t2;p3t; 8E.
Do the two particles collide? If so, at what time do they collide?
Solution: In order for the particles to be in the same place at the same time, there
must be some time tsuch that all of the equations
3t2= 3
p3t=p3t
8 = 6t2+8p312t+ 14 8p3
are satis…ed.
The …rst of these equations is satis…ed for t= 1 and so is the second. Let us see if
t= 1 satis…es the third equation:
6 (1)2+8p312(1) + 14 8p3 = 6 + 8p312 + 14 8p3 = 8.
Thus the particles do collide at time t= 1.
2. Find the unit tangent vector to the curve
r(t) = D3t2;p3t; 8E
at the point on the curve corresponding to t= 1.
1
