Answer: When a ball is dropped into sand, a circular crater is formed due to the impact of the ball.
The size of the crater varies with the kinetic energy of the ball at impact. In this research,
the relationship between the kinetic energy of a modified table tennis ball, dropped into
sand from a fixed height, and the diameter of the crater formed will be investigated.
The size of a meteor crater is related to the amount of energy the meteor possesses as it
strikes the ground. The more energy it possess, the larger the crater diameter. According
to the Palaeontology Department, University of Bristol website, the relationship between
the kinetic energy of a meteor at impact and diameter of the resulting crater on earth is:
4.3/1)(07.0
t
a ECfD
ρ
ρ = ⋅ ,
where D is the crater diameter, Cf is the crater collapse factor, ρa is the density of the
projectile, ρt is the density of the target rock, and E is the total energy of the meteor.
(http://palaeo.gly.bris.ac.uk/communication/Brana/impact.html)
Equation 1 describes craters formed by meteors that collide at high speed with the surface
of the earth. Given that Cf and ρt are constant (and unknown) in our research, and that
both E and ρa are proportional to the mass, m, of our projectile, equation 1 can be
simplified to the form:
4.3/1 ⋅= mEkD )( (2)
where k and n are unknown constants. For the purpose of our research, E is defined as
the kinetic energy at the instant the ball impacts the sand, and since the ball is being
dropped from constant height, it is assumed that the ball is impacting with constant
velocity. It follows that:
59.0
59.0
4.3/2
4.3/12 )5.0(
D k E
ED
ED
mvmkD
⋅=
∝
∝
⋅⋅⋅=
According to equation 3, it is predicted that the diameter of a meteor crater is
proportional to the energy of the meteor raised to the power of 0.59. It is not known if
the relationship based on meteor craters given in equation 3 will be applicable to our
situation, which involves table tennis balls filled with lead shot and latex glue dropped
into sand at low speeds. It is expected that the relationship between the kinetic energy of
the projectile and the resulting crater diameter will be a power law, but with exponent
different from that predicted by equation 3.
This research will attempt to determine the relationship between the kinetic energy at
impact of a modified table tennis ball dropped from a fixed height into sand, and the
diameter of the crater formed. It will also be determined whether it is the same as the
relationship for meteor craters.
Seven table tennis balls of the same volume and different masses were prepared as
follows. Tiny holes were drilled in each ball to insert lead shot and glue. Different
amounts of lead shot were put in to each ball to vary the mass of the balls. Latex glue
mixed with a small amount of water was inserted through the hole of the ball using a
syringe until the ball was completely filled. Glue was used to spread the lead shots evenly
inside the ball to make its density uniform. After filling each ball with glue and lead shot,
clay was used to cover the hole. A minimal amount of clay was used in order not to
change the spherical shape of the balls. The balls were massed using an electronic mass
balance.
As seen above in figure 1, a basin filled with a large amount of sand was prepared. A
circular basin was prepared so that any wall effect stays constant during the research. The
sand was filtered using a sieve with small holes. After that the sand was poured into a
basin which was placed next to a table. On the table, a stand was placed. As seen in
figure 2, three lengths were measured; L1 (top of stand to the floor), L2 (height of basin),
and L3 (top of basin to sand).
Before dropping the balls, the sand was loosened by mixing the sand with the hands. The
sand was flattened by running a straight ruler across the top of the loosened sand. Each
ball was dropped from the top of the stand (see figure 1). The ball was dropped using two
fingers. Each finger was released simultaneously so that the ball did not spin as it fell.
After the ball impacted the sand, the diameter of the crater was measured using a ruler.
The diameter of the outer edge of the crater was measured as seen in figure 3.
After measuring the diameter of
the crater, the sand was loosened
again and the same procedure
was repeated. For each ball, a
total of 5 trials was conducted to
quantify random errors.
Throughout the experiment, the
same type of sand was used, and
the whole experiment was
conducted in the laboratory room
in one hour.
Step-by-step explanation: There is more to tell but this is all for now
, where the vertex (h, k), is (-9, 0).