Fall 2009
Week 5: Projectile motion
their assumptions were not false, and therefore their conclusions were absolutely correct.
Reading Assignment
Galilei, G. 1998, Dialogues Concerning Two New Sciences, Prometheus Books, Amherst, New York, Sec. 268293.
Vocabulary words
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
perpetual
propensity
semi-parabola
conic section
right cone
apex
abstract
fallacious
quadrature
steelyard
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
minute of arc
great circle
plumbline
perturbation
impedance
cubit
frequency of vibration
ordnance
momentum
rectilinear
Introductory comments on Galileo’s (lack of) use of units and formulae:
Galileo scrupulously avoids writing formulae as we do today, such as mass = density times volume. Instead, he relates ratios of like quantities. For example, he describes the law of the lever by saying that in
equilibrium, the ratio of the lengths of the two balancing lever arms is the same as the inverse ratio of the
weights applied. Why doesn’t he just write and do math like we write and do math today?
He writes in this way because it was not clear to natural philosophers of the time that one could multiply
or divide “unlike” quantities. This is not as ridiculous as it may seem. For example, is it really sensible to
multiply four apples by three oranges? Or is it sensible to multiply four pounds by three feet? This is exactly what we do today: we willy-nilly cross out and compound different units to simplify our equations.
In the time of Galileo, the natural philosophers were not so cavalier in their calculations. By relating only
ratios of quantities with identical units, the question of the validity of multiplying or dividing unlike quantities was avoided entirely. Units never appeared in formulae. Furthermore, the relative size of quantities
could be likened to the relative lengths of line segments, and all calculations could be accomplished using
geometrical relationships between lengths of line segments. Every step in a calculation therefore had a
visualizable, geometrical interpretation.
Homework exercises
1.
Geometry problem Consider Fig. 106. Prove that “the square of fe is equal to the rectangle formed by
ge and eh.” (Euclid would say that the mean proportion of ge and eh is fe.) Hint: consider a triangle
inscribed in a circle whose diameter is ge plus eh and whose altitude is fe.
General Physics I - Week 5 1
Fall 2009
Week 5: Projectile Motion
2.
Projectile trajectory problem Consider Fig. 108. Suppose that the line bc has a length of 300 feet and
that the time interval between points b and c is one second. What are the lengths of the lines hl, ci,
and eh?
3.
Projectile motion essay Describe three objections raised against Salviati’s assertion that the motion of
a projectile is described by a (semi)parabola. How does Salviati counter these objections? Are his
refutations reasonable?
4.
Projectile speed problem Consider Fig. 112. Our ultimate goal is to determine the speed of the projectile at points c and e. We will follow Galileo’s geometrical method for determining the speeds.
Note: If you find it strange that Galileo often uses the same line segment to represent both a distance
and a time, recognize that you often do the same thing. For example, you use the same number, six, to
represent both an amount of time (6 seconds), and also a distance (6 feet). Galileo could have drawn
two separate diagrams, one for distances and one for time, but he chose to just reuse the same line
segments in the one diagram to represent different types of quantities.
a.
Galileo says that the length of segment ab represents both a distance and a time. If it represents a
distance of 20 meters, what amount of time does it represent? (Hint: you will need to know the
acceleration of gravity.)
b.
Galileo says that the length of segment ab represents both a time and a momentum. What is the
value of the momentum (or speed) that it represents. (Hint: You will need to know the numerical
value of the acceleration of gravity to answer this.)
c.
Using your answer to the previous question, what is the distance represented by the length of the
line segment dc?
d.
What is the distance represented by the length of the line segment bd?
e.
Which line segments represents the vertical and horizontal momenta at point c? What are the
momenta represented by the length of these line segments?
f.
Which line segment represent the momentum at point c? What is the momentum represented by
the length of this segment?
g.
Suppose the length of segment bf represents a distance of 5 meters. How much time does it take
the projectile to arrive at point e?
h.
If bd provides a measure of the time taken to fall through distance bd, and bg provides a measure
of time taken to fall through distance bf, then what is the ratio of the segments bg and bd? What is
the ratio of the segments bf and bd?
i.
Show that bg is the mean proportion of bf and bd.
j.
Which line segment represent the vertical momentum acquired at point e? What is the momentum represented by the length of this line segment?
k.
Which line segment represents the horizontal momentum at point e? What is the momentum
represented by the length of this line segment?
General Physics I - Week 5 2
Fall 2009
Week 5: Projectile Motion
l.
Which line segment represent the momentum at point e? What is the momentum represented by
the length of this segment?
m. What angle represents the direction of motion of the projectile at point e? What is the numerical
value of this angle?
5.
Archery problem An archer fires an arrow over level ground at an angle of 75 degrees above the horizontal with an initial speed of 35 m/s. You may use whatever method you choose to solve this problem.
a.
What is the maximum height achieved by the arrow?
b.
What is its time of flight?
c.
What is its range?
Laboratory Exercises
The purpose of this laboratory exercise is to determine the horizontal range of a cannon ball launched at a
known angle with respect to the horizon. You will utilize a small spring loaded cannon, small steel cannonballs, a sheet of carbon paper and a meter(yard)stick. Be sure to record all data neatly in tables in your
notebook and make plots where appropriate to illustrate your data.
A. In order to do range predictions, you must first determine the muzzle velocity of the cannon. You can
do this by launching a ball horizontally from a known height above the floor and measuring the horizontal distance to the point of impact. You will need to tape a sheet of paper to the floor and place
carbon paper over the paper. This will allow you to identify precisely the location of the impact. Be as
precise as possible in your measurements, since your later work will depend critically upon your value
of the muzzle velocity of the cannon. To this end, you might wish to fire five shots and find the average of the distances. How does this measurement tell you the horizontal speed of the cannonball, and
hence the muzzle velocity? (Hint: you might consider separately the horizontal and the vertical motion of the ball. How much time does the ball take to strike the ground from a given height? And
how far does it travel horizontally during this time?)
B.
Now, using the known muzzle velocity, try to predict the maximum height which the ball can reach if
aimed straight upward. Get your laboratory instructor to watch, and then test your prediction by aiming the projectile launcher vertically and holding a meter stick up. Be sure to be as quantitative as
possible in your analysis.
C.
Next, set up the cannon to fire at an angle of 25 degrees above the horizon onto the floor below. Before firing your cannon, make an exact prediction of the range using your muzzle velocity and the
known vertical distance to the floor. (Hint: you will need to determine the vertical component of the
velocity and then determine the total time that the projectile is in the air before determining the horizontal range.) Now, draw a horizontal line on a sheet of paper fixed to the floor and place carbon paper over it. When you are confident of your prediction, get the instructor to watch your test firing.
The proximity of your impact to the predicted range is a measure of your success.
D. Finally, experimentally determine the angle at which a cannonball should be shot so as to maximize
the horizontal range. Is it what you would expect?
General Physics I - Week 5 3