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Gravity and 2-D Ballistic Motion 1 Introduction This lab focuses on ballistic motion, or the path of a body moving only under the
influence of gravity. In particular, we will measure the value of gravitational
acceleration and analyze the trajectories of launched projectiles. Newton’s laws are
important here, since they provide the theoretical foundation for describing such motion
mathematically. This experiment is intended to take two class periods, and you will
write a full laboratory report for it. For guidance, refer to the file on eLearning called
Writing Lab Reports, which has a description of all the information you will need to
include. Remember that this lab report is 25% of your final grade. 2 Key Concepts •
• Projectile motion • Newton’s laws • 3 Gravitational acceleration Full laboratory write-up Theory 31 Newton’s Laws 1. Law of Inertia: Every object in uniform motion—that is, moving with
constant velocity—will stay in uniform motion unless a net external
force acts on it.
~
2. Fnet = m ~a. The net force acting on an object equals the mass of the
object times its acceleration. Remember that force and acceleration are
vector quantities, while mass is a scalar. 3. When one object exerts a force on a second object, the second object
also exerts a force on the first object that is equal in magnitude and
opposite in direction. This is often phrased as “for every action, there
is an equal and opposite reaction.” The second law is most useful to us in this lab, because we know that the force of
gravity is, to very good approximation, constant and pointing downward near the
surface of the earth. The second law says that a constant force acting on an object
translates into constant acceleration of the object, and under the assumption of constant
acceleration, the following kinematic equations can be derived. See your text or the
appendix of Lab 3 for more details. 1 ~x = ~v0t ~a
+
1t2 , (1) 2
~v = ~v0 + ~a t , (2) ~v2 = ~v02 + 2~a ·
~x . (3) Time t is the independent variable, ~x is the change in position, ~v 0 is the initial
velocity, ~v is the final velocity (at time t), and ~a is the acceleration. Note that in the
third equation, there is a dot product between ~a and ~x, and the notation ~v 2 means ~v
· ~v, the square of the magnitude of ~v.
The equations above are vector equations, which we can think of as describing
relationships be-tween geometrical arrows associated with our object moving in threedimensional space. However, it is often more useful to use the equations in component
form relative to a Cartesian coordinate
ˆˆ ˆ system (the usual ı, , and k), although other coordinate systems are possible. This means
that Equations (1) and (2) each really stand for three equations—one for each coordinate
of the motion. For example, from Equation (2), we have v x = v0x +axt, and likewise for y
and z. We can therefore treat the motion along different coordinate directions separately
in our analysis when the motion occurs in more than one dimension. Making a table of
the components of each variable is a helpful tool in keeping track of things. Consider
Figure 1 and Table 1 for a 2-D example of this process. Figure 1: Diagram of general projectile motion. 32 Projectile Motion We will be using the kinematic equations above to analyze the trajectory of a launched
projectile. Projectile motion occurs when an object is subject to an initial force that
propels it into the air, after which it follows an arcing path to the ground. In reality,
projectile motion is complicated by the presence of air resistance and other external
forces besides gravity. However, in this lab, we will neglect these other influences, and
our equations will only reflect the effect of gravity on the motion. It is important to
remember that we are applying the kinematic equations to the object only when it is
actually in flight; the launcher just serves to give the projectile an initial velocity, and
what happens inside the launcher is irrelevant to the object’s trajectory. 2 ~x (m) ~v0 (m/s) ~v (m/s) Motion description up v max − hi v down v x x 0x ~a
(m/s2) t (s) 0x 0 t 0 −9.80 t fx = v0x 0 t down fy −9.80 t down v up Motion from hi to hmax
y h x x 0y 0x v up Motion from hmax to hf
y f − hmax h 0 v Table 1: Table for organizing components of constant acceleration motion problems.
Have a set of x and y rows for each part of the motion, keeping in mind where the
beginning and ending points of the motion are. This table is set up for a projectile where
you want to know the maximum height (hmax) and range (xup + xdown). Calculations in trajectory problems can be lengthy, as there are many quantities of
interest. The maximum height, the total flight time, the total horizontal range, and the
initial and final velocities and angles are all variables in this type of problem that you
might be asked to solve for. When solving these equations, keep in mind the following
key points. 1. The projectile attains its maximum height when the vertical component of its
velocity is zero.
2. The acceleration due to gravity points vertically downward and has magnitude
9.80 m/s2.
3. All objects free fall at the same rate in the absence of air resistance.
4. Since time is a scalar, the time in the x direction is the same as the time in the y direction.
5. Be careful to take into account the difference in height between your initial
and final positions. For more information on Newton’s laws and projectile motion, see the appropriate
sections in your text. For more information on analyzing graphs, see Lab 3 and the file
called Graphing Skills on eLearning. 33 Full Laboratory Report A full laboratory report is a professional way to present all of the data