Flying Through the Air
Instructors’ Notes
Introduction
This is a unique lab. First, it is the first time students will be doing an
“experiment problem.” That is, they will be asked to solve a physics problem
that is like a textbook problem, but one in which the data must be found by them.
Second, the product of the lab is the theoretical solution to the problem and
the result of a single experimental demonstration. Basically, if they succeed
they get points; if they don’t, they don’t.
The problem is to determine the horizontal distance a projectile travels when it
is launched at a given angle and lands at an elevation different from the level
at which it was launched.
They will then need to do three things. First, determine the initial speed of
the ball. Second, test their theoretical equation to see if it works for a
variety of angles. Third, figure out the uncertainty in the distance by seeing
the variation in distance for repeated launches. The third test is important
because it will tell them whether they should “chicken out” and accept 4 points
for landing within 4 cm of the target (8 points total), try the reasonable
uncertainty of 2 cm and get 20 points, or go for 4 extra credit by aiming for a 1
cm uncertainty. This exercise should drive home both the reason why they have to
pay attention to uncertainties and motivate them to develop techniques that
produce reproducible results. Those, friends, are my “hidden agendas.”
Grading Guidelines
Split the 12 points for the theoretical and experimental preparation equally. I
would suggest two for the pictorial and physical models and four for the
mathematical. Take off if they didn’t do the early parts. Their derivation,
which is really most of the pre-lab exercise, should have been at least partially
done before coming to class. I would suggest splitting the six experimental
points among the measurement of the initial velocity (2), some note on
approximations used (1), measurements of distance versus angle with comparisons
to theory and an indication of the spread in hits.
Pre-lab Questions
An object has an initial (unknown) velocity v0 . It moves with constant acceleration a. After moving a
distance h its velocity is zero. Write an equation giving the initial velocity in terms of a and h.
v2 = v02 + 2a(x - x0) or v02 = –2ah, v0
Physics Laboratory Instructors’ Manual. Jan 1999
= - 2ah (a = –g)
Flying Through the Air - Page 1 of 6
Do the Theoretical Analysis on page 3. First, do the pictorial and physical models.
Here is a step-by-step approach to the mathematical model.
Strategy
Horizontal (x)
Vertical (y)
Identify the initial position
x0 = 0
y0 = h
Resolve initial velocity v0 at θ0 into x- and
y-components
vx0 = v0 cos θ0
vy0 =
Using your coordinate system, identify
components of acceleration.
ax = 0
ay = –g
Specialize generic equation
x = x0 + v0 t + ½ at2 for x- and y-directions
x = v0 cos θ0 t
y = h+v0 sin θ0t – ½gt2
Identify location where projectile lands
x=L
y=0
Find time when it lands in terms of L and
vx0 and substitute into equation for y.
t =L/v0 cos θ0
0 = h +v0sinθ0L/v0cosθ0 –
Equation for y position is now a quadratic
equation involving L. Solve it for L.
Your equation will involve g, h, L, v0,
tan θ0, and cos θ0.
Physics Laboratory Manual. Sept. 1995
v0 sin θ0
½g(L/v0 cos θ0)2
0 = – h –tan θ0L + (g/2v02 cos2θ0)L2
L=

v0 cos 
2
2
v0 sin  + v0 sin  + 2gh
g

Introduction to the Physics Laboratory - Page 2 of 8
Theoretical Analysis
A projectile is launched at a given speed, v0, and angle, θ0, from a height h > 0 above the ground.
It lands on the ground a horizontal distance L from the launching point. Determine an expression
for the distance L in terms of the initial height, speed, and angle.
Pictorial Model
·
·
·
·
·
Draw coordinate axes.
Sketch the motion.
Identify x and y components of quantities
at key positions during the motion. Start: x0=0, y0 = h.
List known constants. a = -g = -9.80 m/s2
Identify unknowns that need to be
determined. v0
v0x = v0 cos θ0, v0y = v0 sin θ0
Physics Laboratory Manual. Sept. 1995
End: x = L, y = 0
Introduction to the Physics Laboratory - Page 3 of 8