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Impact crater experiments for introductory physics and astronomy laboratories J R Claycomb Department of Mathematics and Physics, Houston Baptist University, Houston, TX 77074, USA E-mail:
[email protected]
Abstract Activity-based collisional analysis is developed for introductory physics and astronomy laboratory experiments. Crushable floral foam is used to investigate the physics of projectiles undergoing completely inelastic collisions with a low-density solid forming impact craters. Simple drop experiments enable determination of the average acceleration, force, collision time, impulse, and work done on simulated meteorites by the floral foam during crater formation.
Introduction
Materials and methods
A meteorite impacting an astronomical body will likely vaporize upon impact, resulting in a spherical crater even for grazing incidence [1]. NASA has performed studies with iron projectiles impacting low-density porous floral foam simulating comet nuclei [2]. NASA flyby spacecraft observed a spectacular example of this type of collision during the 2005 deep impact mission [3], in which a 370 kg copper-core smart impactor targeted comet Temple 1. Smaller scale inelastic collisions were encountered during the Stardust sample return mission [4], in which dust particles from comet Wild 2 were collected. These micron-size particles were captured in Aerogel foam blocks on board the Stardust spacecraft and returned to Earth in 2006. The impact velocities encountered in laboratory experiments reported here are much smaller than the velocities associated with astronomical collisions. However, relevant physical principles may be investigated in the context of astronomical phenomena that capture students’ imaginations.
Laboratory materials include crushable floral foam blocks (available at craft stores [5] or floral shops), metallic spheres, a mass balance, callipers with a depth measurement rod, a 2 m stick, and a flat plastic ruler. The more pliable floral foam has a dark green colour compared to the stiffer light green foam that would be less suitable for these experiments. As a practical point, it is very compelling to dig one’s fingers into the soft floral foam. Students can inadvertently degrade the foam if not cautioned beforehand. Also, one should try to select foam from the craft or floral shop that is not riddled with finger and thumb impressions. The masses of the metallic spheres are first determined using a triple beam balance. A 2 m stick is positioned above the foam target, as demonstrated in figure 1. The masses, m , are released from various heights above the target. For greater accuracy, drops at a given height may be repeated several times and the crater depths averaged. The depth measurement rod of the
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Impact crater experiments for introductory physics and astronomy laboratories
Figure 2. Physics student Edith Gonzalez measuring the impact crater depth using a ruler and vernier callipers. Figure 1. Physics student Edith Gonzalez (left) and Society of Physics Students (SPS) President Suhare Adams (right) releasing a metallic sphere above the floral foam block from measured drop heights.
0.035
vernier callipers is used to measure the crater depth after each impact. This procedure is demonstrated in figure 2: a flat plastic ruler is placed over the crater, spanning its full diameter, to stabilize the callipers while the depth gauge is inserted. Figure 3 shows a plot of crater depths as a function of drop heights with a (12 mm radius, 66.4 g) brass sphere released from 5– 200 cm above the target, in increments of 5 cm. The linear variation shows that the crater depth is proportional to the impact energy, equal to mgh drop. This impact energy can be used to calculate the speed of the projectile just before impact by applying energy conservation: 2 1 mvimpact 2
giving the impact speed
vimpact =
= mgh drop,
(1)
2gh drop .
(2)
From the measured crater depth and the calculated velocity before impact, the average acceleration of the ball during each impact is 2 /2δ , or calculated as aavg = −vinpact
aavg March 2009
h drop , = −g δ
(3)
crater depth (m)
0.030 0.025 0.020 0.015 0.010 0.005 0.000 0.0
1.5 0.5 1.0 drop height (m)
2.0
Figure 3. Measured crater depth versus drop height.
where δ is the depth of the crater. Figure 4 shows a plot of aavg versus drop height. Very large negative accelerations between 10g and 60g are calculated for drop heights up to 2 m. The average acceleration and the calculated impact velocity are used to calculate the collision time Tcol = −vimpact /aavg , or 2 Tcol = δ . (4) gh drop Values of T in the range 8–10 ms are plotted as a function of drop height in figure 5. The average acceleration is also used to calculate the PHYSICS EDUCATION
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J R Claycomb Table 1. Collisional quantities with the simulated meteorite released from various heights above the floral foam. Work quantities in brackets are obtained using equation (6). Drop height (m)
Impact speed (m s−1 )
Impact Crater energy (J) depth (m)
0.25 1.00 2.00
2.22 4.43 6.26
0.163 0.651 1.3
Average acceleration (m s −2 )
−285 −481 −613
0.0086 0.0204 0.032
–200
collision time (s)
acceleration (ms–2)
–100
–300 –400 –500 –600 –700 0.0
0.5 1.0 1.5 drop height (m)
2.0
(5)
(6)
More accurately, including the work done by gravity during the collision, we have (7)
The impulse, or change in momentum of the ball, is calculated with the final velocity of the ball equal to zero:
I = p = −m ball vimpact = Favg T.
(8)
Table 1 records the measured drop height and crater depth, along with the calculated impact speed, impact energy, acceleration, average force, collision time, work done on the projectile, and the impulse. Data are illustrated for three different drop heights in this shortened table. 186
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18.9 31.9 40.7
0.168 (0.163) 0.007 77 0.147 0.665 (0.651) 0.009 21 0.294 1.32 (1.302) 0.010 2 0.416
0.0110 0.0105 0.0100 0.0095 0.0090 0.0085 0.0080 0.0075 1.5 0.5 1.0 drop height (m)
2.0
Extensions and discussion
and hence the work done by the floral foam on the ball may be estimated:
Wfb = mg h drop + δ .
Collision Impulse time (s) (Ns)
Figure 5. Impact collision time versus drop height.
average force imparted on the ball,
Wfb = Favg δ.
Work (J)
0.0
Figure 4. Acceleration of colliding sphere during impact versus drop height.
Favg = m ball aavg ,
Average force (N)
Alternative measurements of crater depth Silicone castings of the crater can be made, and the resulting crater depth measured from the casting height. This procedure may not be practical for shorter laboratory sessions with drying times of the order of hours. The crater depth can also be determined from the measured diameter 2r for crater depths less than the sphere radius R as shown in figure 6, where δ is calculated from
(R − δ)2 + r 2 = R 2 .
(9)
Splash craters Floral foam is highly absorbent to liquids. Wet foam may be impacted to simulate splash craters such as those observed on Mars [6]. Splash crater experiments could be conducted as a laboratory extension experiment after data are collected with dry foam. Discussion of conservation of momentum This laboratory experiment provides opportunities for guided discussions of the conservation of March 2009
Impact crater experiments for introductory physics and astronomy laboratories with actual aavg and percentage (aavg /aavg ) × 100% uncertainties. For the average force we have the fractional uncertainty
Favg aavg m ball = + Favg aavg m ball
δ
with actual Favg and percentage (Favg /Favg ) × 100% uncertainties. The fractional collision time uncertainty is given by
R
R-δ
Tcol δ 1 h drop + = Tcol δ 2 h drop
r
Figure 6. Geometry for determining the crater depth from the crater radius for crater depths less than the ball radius.
momentum and energy during the free fall of the sphere and its collision with the floral foam. For example, if the momentum of the sphere is m ball vimpact immediately before impact, how is momentum conserved if the sphere subsequently comes to rest? After posing this question, students can be asked to consider the Earth–sphere system, in which the total momentum is unchanging during free fall and the collision. The acceleration of the Earth during free fall of the sphere can then be calculated from Newton’s third law.
A comparison of the impact crater depths for collisions with the same impact energy but with different impact speeds may be performed with aluminium and brass (or steel) spheres with the same radius. The aluminium sphere must be dropped from a greater height h Al , determined from m Al gh Al = m brass gh brass . (10) Here the average δ from several trials of each sphere should be compared. Discussion of errors Fractional, percentage and actual uncertainties are calculated based on uncertainties in the measured drop height h drop , crater depth δ , and projectile mass m . The fractional uncertainty in average acceleration is given by
March 2009
(13)
with actual Tcol and percentage (Tcol /Tcol ) × 100% uncertainties. The actual uncertainties in each quantity may be compared to standard deviations obtained from multiple trials at a fixed drop height. Additional factors affecting the crater depth can be discussed, such as variable foam density during compactification and the changing surface area between the sphere and the foam for δ < R.
Acknowledgment I gratefully acknowledge helpful suggestions from Victor Lopez in measuring the crater depth using silicone castings and the help of Amitpal Tagore with crater depth measurements. Received 11 October 2008, in final form 19 December 2008 doi:10.1088/0031-9120/44/2/011
Varying collision speed with constant energy
h drop δ aavg = + aavg h drop δ
(12)
(11)
References [1] Seeds M A 2008 Exploring the Universe 10th edn (Belmont, CA: Thomson) p 380 [2] Durda D D, Flynn G J and Van Veghten T W 2003 Impacts into porous foam targets: possible implications for the disruption of comet nuclei Icarus 163 504–7 [3] Warner E M and Redfern G 2005 Deep impact: our first look inside a comet Sky Telesc. 1 40–5 [4] Kissel J, Krueger F R, Sil´en J and Clark B C 2004 The cometary and interstellar dust analyzer at Comet 81P/Wild 2 Science 304 1774–6 [5] Wall Mart and Michaels stores, http://www. save-on-crafts.com/floraltoolbox.html [6] Seeds M A 2008 Exploring the Universe 10th edn (Belmont, CA: Thomson) p 400
J R Claycomb is currently an assistant professor in physics at Houston Baptist University. He teaches modern physics, biophysics and astronomy, and his research interests include biophysics and magnetic measurements using SQUID magnetometers. PHYSICS EDUCATION
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