Attachment to Oceana's Comment - Vanderlaan & Taggart

MARINE MAMMAL SCIENCE, 23(1): 144–156 (January 2007)

C 2006 by the Society for Marine Mammalogy
DOI: 10.1111/j.1748-7692.2006.00098.x

VESSEL COLLISIONS WITH WHALES:
THE PROBABILITY OF LETHAL INJURY
BASED ON VESSEL SPEED
ANGELIA S. M. VANDERLAAN
CHRISTOPHER T. TAGGART
Department of Oceanography,
Dalhousie University,
Halifax, NS B3H 4J1, Canada
E-mail: [email protected]

ABSTRACT
Historical records demonstrate that the most numerous, per capita, ocean-goingvessel strikes recorded among large-whale species accrue to the North Atlantic
right whale (Eubalaena glacialis). As vessel speed restrictions are being considered
to reduce the likelihood and severity of vessel collisions with right whales, we
present an analysis of the published historical records of vessels striking large whales.
We examine the influence of vessel speed in contributing to either a lethal injury
(defined as killed or severely injured) or a nonlethal injury (defined as minor or no
apparent injury) to a large whale when struck. A logistic regression model fitted
to the observations, and consistent with a bootstrap model, demonstrates that the
greatest rate of change in the probability of a lethal injury (P lethal ) to a large whale
occurs between vessel speeds of 8.6 and 15 knots where P lethal increases from 0.21
to 0.79. The probability of a lethal injury drops below 0.5 at 11.8 knots. Above
15 knots, P lethal asymptotically approaches 1. The uncertainties in the logistic regression estimates are relatively large at relatively low speeds (e.g., at 8 knots the
probability is 0.17 with a 95% CI of 0.03–0.6). The results we provide can be used
to assess the utility of vessel speed limits that are being considered to reduce the
lethality of vessels striking the critically endangered North Atlantic right whale
and other large whales that are frequent victims of vessel strikes.
Key words: vessel strike, vessel speed, lethal injury, whales, right whale, probability,
logistic regression, bootstrap.

Recently compiled historical (1885 through 2002) records of vessels striking large
whales worldwide (n = 294; Laist et al. 2001, Jensen and Silber 2003) reveal the
most frequently reported victims of vessel strikes to be fin (Balaenoptera physalus),
humpback (Megaptera novaeangliae), North Atlantic (NA) right (Eubalaena glacialis),
gray (Eschrichtius robustus), and several other large whales (Fig. 1). On a per-capita basis
using contemporary worldwide population-size estimates (Aguilar 2002, Clapham
2002, Ford 2002, Horwood 2002, Jones and Swartz 2002, Kato 2002, Kenney 2002,
144

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VANDERLAAN AND TAGGART: VESSEL COLLISIONS WITH WHALES

–2

70
–3

Number of whales struck

10
60

50

–4

10

40
–5

10

30

20
–6

10
10

Proportion struck per capita per year

10

80

–7

10

0
fin

humpback NAR

gray

minke

SR

sperm

blue

sei

Bryde’s

orca

unknown

Whale species (common name)

Figure 1. Frequency histograms of worldwide documented (Laist et al. 2001, Jensen and
Silber 2003) numbers of large whales, including North Atlantic (NAR) and southern (SR)
right whales, reported struck by vessels for the period 1960 though 2002 only (open bars),
and the same data presented as a temporally adjusted per capita rate (solid bars; log 10 scale)
using contemporary population size estimates for each species (Aguilar 2002, Clapham 2002,
Ford 2002, Horwood 2002, Jones and Swartz 2002, Kato 2002, Kenney 2002, Perrin and
Brownell 2002, Sears 2002, Whitehead 2002) where the proportion struck per capita per year
= (number of species-specific whales struck/contemporary species-specific population size)/43
years. Where a range in population size was provided, we use the midpoint of the range.

Perrin and Brownell 2002, Sears 2002, Whitehead 2002), and relative to all other
large whales reported struck over the period 1960–2002 inclusive (n = 275), the
NA right whale is two orders of magnitude more prevalent as victim (Fig. 1). These
statistics suggest that relative to other large whales, NA right whales are more prone
to being struck by vessels.
Following the U.S. National Oceanic and Atmospheric Administration (NOAA)
advance notice of proposed rulemaking (Federal Register (USA) 2004) for right whale
ship-strike reduction, Kraus et al. (2005) called for emergency measures to reduce
ocean-going vessel speeds in east-coast regions of the United States and thereby to
reduce vessel-related NA right whale mortality. The call for emergency measures
rested on arguments that (1) the NA right whale is the most endangered species of
baleen whale (Kraus et al. 2001); (2) the population size is diminishing (Fujiwara and
Caswell 2001); (3) species extinction is expected within ∼200 years unless humaninduced kills are reduced (Caswell et al. 1999); (4) of all documented kills, most
are attributable to vessel-strike (Knowlton and Kraus 2001); and (5) contemporary
vessel-kill rates remain high (Kraus et al. 2005). Subsequently, and in an attempt to
reduce mortalities due to vessel strikes, the NOAA proposed rule (Federal Register
(USA) 2006) aims to “impose vessel speed restrictions of 10 knots or less” in “certain
areas and at certain times of the year, or under certain conditions,” and “also invites
comments on vessel speed restrictions of 12 knots or less, and 14 knots or less.”

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The above observations, arguments, and proposals led us to estimate the probability
of a lethal injury (i.e., killed or severely injured) to a large whale as a function of vessel
speed at the time of the vessel–whale collision. We report statistically determined
estimates of the probability of a lethal injury and their associated 95% confidence
intervals (CIs) based on vessel speed and offer the estimates as a first step toward
assessing the utility of vessel speed restrictions in areas where vessels are likely to
encounter whales.
METHODS
We use the only published sources detailing the historical record of vessels striking
large whales (n = 294; Laist et al. 2001, Jensen and Silber 2003) where the records (n =
47) jointly provide the vessel speed estimate and the severity of injury to the stricken
whale. Laist et al. (2001) describe four injury classes: killed (carcass observed); severe
(bleeding wounds and/or blood in the water); minor (visible nonbleeding wound,
signs of distress, no report of blood); none apparent (resighted, no visible wound or
distress, animal resumed prestrike activity); and a 5th unknown-injury class (animal
not observed again and no report of blood). Jensen and Silber (2003) assess injury
differently, though their descriptions allowed us to classify according to the four
injury classes of Laist et al. (2001). Those data where speed was known and injury was
unknown are excluded. “Unknown” species are included where speed was known.
Apart from the unknown species, all but one record (Orcinus orca, retained) involved
large whale species (Fig. 1). We use knots as the unit of speed as it is the nautical
convention. Vessel speed is classified in two-knot intervals for all analyses except
in the chi-square tests described below. If vessel speed was reported as a range, the
midpoint is used. One case reported <10 knots for a vessel accelerating at the time
of strike and a speed of 10 knots is assumed for the analyses.
The few data detailing vessel collisions with right whales require us to assume
that the other large whales, primarily baleen whale species (Fig. 1), serve as suitable
proxies, at least from a body-mass perspective. This assumption is justified by the
average mass at maximum length relation provided by Trites and Pauly (1998) that
shows one relation applies to all mysticetes and sperm whales (Physeter macrocephalus)
with a mid point mass of 42.5 × 103 kg. Additionally, the average mass across
all species, excluding Eubalaena sp. and O. orca, based on data provided by Lockyer
(1976), is 39 × 103 kg (n = 219, CV = 84%), and the mean of the species-specific
means is 31 × 103 kg. These estimates above are broadly consistent with the 39 ×
103 kg estimate for a 20 year-old right whale (Moore et al. 2004).
Chi-square tests are used to assess the independence of vessel speed and the severity
of injury according to the four injury-classes of Laist et al. (2001) above. We employ
the simple logistic regression model, Plethal = 1+exp−(10 +1 speed) , (e.g., Myers et al. 2002)
using mid point speeds among the two-knot speed classes, the proportion of whales
suffering either “nonlethal” or “lethal” injury, and maximum likelihood estimation to
determine the parameters and the CIs around model estimates. We define nonlethal
as the sum of the minor and none-apparent injury classes above, and lethal as the sum
of the killed and severe injury classes above. In the latter case, we explicitly assume
a severely injured whale ultimately succumbs to the injury. This assumption has
some merit for a number of reasons (1) other evidence of vessel strikes, such as scars
from propeller wounds on live animals, has a low incidence of reporting (7%) and is
interpreted as indicating such strikes are deadly to NA right whales (Kraus 1990);

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0.50
Killed

Severe

Minor

None

0.45

Cummulative per cent

0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
0

2

4

6

8

10

12

14

16

18

20

22

24

Vessel speed (knots)
Figure 2. Cumulative per cent increase in each of four whale-injury classes as a function of
the midpoint of the two-knot vessel-speed classes illustrating how the killed and severe injury
classes increase similarly and in parallel, as do the minor and none apparent injury classes.

(2) of the documented vessel strikes in the NA right whale population, 1/2 of known
propeller injuries proved fatal (Knowlton and Kraus 2001); (3) blunt trauma that is
consistent with vessel strike is not externally obvious and frequently results in death
(Wiley et al. 1995, Best et al. 2001, Moore et al. 2004); and (4) the cumulative percent
of the killed and severe-injury classes in the data we examine increase similarly and
in parallel with speed as do the cumulative percent minor and none-apparent injury
classes, though the latter at a lower level (Fig. 2). The results of the logistic regression
are used to draw inferences based on its inflection as well as on the two inflections of
the first derivative of the functional relation.
The reliability of the data and the simple logistic regression model are examined
using a bootstrap technique computed using “R” (R Development Core Team 2005)
by resampling the data, with replacement, 1,000 times and by fitting the logistic
to the resultant predicted probability distributions (based on nonlinear least squares
estimation) across speed classes.
RESULTS
Speed and injury are not independent (6 df, P = 0.014) when vessel speed is
categorized across three 8-knot speed intervals: low (0 ≤ knots ≤ 8), moderate (8 <
knots ≤ 16), and high (>16 knots); that is, as speed increases the severity of injury

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0.6
0.4
0.2

Probability of lethal injury

0.8

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MARINE MAMMAL SCIENCE, VOL. 23, NO. 1, 2007

0.0

post-publication Fig. 3
with each datum posted
2

4

6

8

10

12

14

16

18

20

22

24

Vessel speed (knots)

Figure 3. Probability of a lethal injury resulting from a vessel strike to a large whale as a
function of vessel speed based on the simple logistic regression (solid heavy line) and 95% CI
(solid thin lines) and the logistic fitted to the bootstrapped predicted probability distributions
(heavy dashed line) and 95% CI for each distribution (vertical dashed line) where each datum
() is the proportion of whales killed or severely injured (i.e., lethal injury) when struck by
a vessel navigating within a given two-knot speed class. There are no data in the 4–6 knot
speed class.

increases. The same test based on four-speed classes incrementing at six knots and
three-speed classes incrementing at 10 knots, and assessed against the four severitiesof-injury, leads to the same conclusion (9 df, P = 0.0007 and 6 df, P = 0.0001,
respectively).
The probability of a lethal injury (Fig. 3) as a function of vessel speed (knots) is
1
. Wald’s chi-square shows both  0 and  1
determined as: Plethal = 1+exp−(−4.89+0.41speed)
as different from zero (P = 0.013 and 0.003, respectively), and the overall model is
significant (P < 0.001) according to a likelihood ratio test. The logistic fitted to the
bootstrapped probability distributions has similar parameter estimates:  0 = −5.76
and  1 = 0.51.
The simple logistic regression model (Fig. 3) shows that the greatest rate of change
in the probability of a lethal injury to a large whale occurs between 8.6 knots

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Table 1. The odds ratio and associated lower and upper 95% confidence limit of a lethal
injury to a large whale occurring at a given vessel-speed increment.
Speed increment (knots)
1
2
3
4
5

Odds ratio

Lower 95% limit

Upper 95% limit

1.51
2.29
3.45
5.22
7.89

1.15
1.32
1.52
1.75
2.02

1.99
3.94
7.83
15.5
30.9

(P lethal = 0.21) as defined by the first inflection of the first derivative of the logistic
and 15 knots (P lethal = 0.79) as defined by the second inflection of the first derivative.
Only at speeds below 11.8 knots (inflection of the logistic) does the probability of
a lethal injury drop below 0.5, though the uncertainties around the estimates are
large. Above 15 knots P lethal asymptotically approaches 1. The odds ratio, that is the
Plethal
, of a lethal injury occurring at a given initial speed relative
ratio of the odds, 1−P
lethal
to the odds at some incremented speed, increases with the magnitude of the speed
increment (Table 1). For example, an increase in vessel speed by 1 knot increases
the odds of a lethal injury 1.5-fold (95% CI 1.2 2.0) regardless of initial speed. A
two-knot increase in speed increases the odds by 2.3-fold (95% CI 1.3 3.9) and a
five-knot increase leads to a 7.9-fold (95% CI 2.0 31) increase in the odds of a lethal
injury.
The logistic fitted to the bootstrapped (with resampling) predicted probability
distributions provides statistically similar results (Fig. 3), and there is no difference in the predicted values derived from the logistic fitted to the bootstrapped
probability distributions and those provided by the simple logistic regression model
(bootstrapped parameters are well within ± 1 SE of the simple logistic parameter
estimates). For this reason the inferences below rely on estimates derived from the
simple logistic regression model and the associated 95% CI.

DISCUSSION
The logistic regression model estimates demonstrate that the greatest rate of change
in the probability of a lethal injury to a large whale, as a function of vessel speed,
occurs between the inflections of the first derivative of the logistic model; that is,
between vessel speeds of 8.6 and 15 knots. Across this speed range, the chances
of a lethal injury decline from approximately 80% at 15 knots to approximately
20% at 8.6 knots. Notably, it is only at speeds below 11.8 knots that the chances of
lethal injury drop below 50% and above 15 knots the chances asymptotically increase
toward 100%.
The data used in our analyses are limited and do not incorporate all variables
(e.g., species of whale, age, size or mass, and behavior; and vessel type, size or mass,
and angle of attack) relevant to vessel–whale collisions. They are, however, the only
published data that include vessel-speed observations. Consequently, the CIs are
large, particularly at low vessel speeds (<10 knots) where there are few observations.
Assuming that the mass of the vessels represented in the data are much greater than
the mass of the whales struck, we conclude that vessel speed is sufficient to predict
the probability of a lethal injury if a whale is struck, where lethality includes killed

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or severely injured. This conclusion is not unreasonable, at least within the limits of
the two extremes of elastic or perfectly inelastic collisions in one dimension and by
assuming that both the mass and speed of the colliding vessel are much greater than
the mass and speed of the colliding whale. In such a simplification, it can be shown that
it is only the mass of the whale and the speed of the vessel that contribute to the impact
forces (see Appendix) and presumably the severity of injury to the whale. Although
this simplification ignores the time over which the collision occurs (t in Appendix)
and how the energy is dissipated during the collision (neither easily determined),
it does demonstrate that vessel speed is expected to be a reasonable predictor of
lethality—at least as a first approximation. It is notable that the functional forms of
the ascending limbs of the logistic models illustrated in Figure 3 are proportional
to the square of the vessel speed and thus consistent with expected collision-related
kinetic-energy dissipation in the whale.
This study provides insights into the role vessel speed plays in determining the
fate of a right whale, or other large whale, if struck. The probability estimates and
their associated 95% CIs provide insight into how effective vessel-speed restrictions
might be in reducing the severity of vessel-strike injuries. Such restrictions may
complement other efforts designed to reduce vessel strikes (Kraus et al. 2005). Despite
increased awareness of the vessel-strike problem and changes to vessel routing, such
as the modified traffic separation scheme in the Bay of Fundy right whale habitat
(International Maritime Organisation 2003), there has not been a reduction in the
reporting of lethal vessel-strike injuries. There were at least three and possibly four
right whale deaths attributed to vessel strikes in the 16 months prior to Kraus
et al. (2005). It is possible that increased awareness may be responsible for increased
reporting. However, if contemporary average vessel speeds of 14–16 knots through
two critical right whale habitats (Ward-Geiger et al. 2005) are maintained, it is
reasonable to expect the probability of lethal vessel-strike injuries to remain in the
0.70–0.85 range based on the simple logistic model (Fig. 3).
One factor our analysis cannot address is the consequence of increased whale exposure to vessels navigating at low speed. Therefore, we briefly explore average vessel–
whale encounter probability (P m ) and how it may change as vessel speed decreases.
We do this by employing a model, in two dimensions, of a random walk (whale), in
the presence of traps (vessels), provided by Gallos and Argyrakis (2001). The probabilities are explored within a specified areal domain, using a vessel frame of reference
and a randomly moving whale with a speed that is the sum of the vessel (v v ) and
the whale (v w ). In this example, and for simplicity, we assume a square domain of
dimension (a) and length (l v ) and beam (b v ) of the vessel and the whale (l w and b w ). To
determine the number of steps in the random walk, we require the time (t v ) for the
vessel to transit the domain and the area (c v ) occupied by the transiting vessel within
the domain. We approximate that, on average, a vessel transit parallels the edges of
the domain; thus, tv = vav . Vessel area within the domain is defined by the number
of vessels (N) and their dimensions: c v = N·lav2·b v . During the time the vessel transits
the domain, the whale will move through an area specified as Aw = b w (v w + v v )t v .
The above equations are used to determine the number of steps (S n ) taken by the
whale during its random walk: S n = lwAbww = law ( vvwv + 1). There are other means of
deriving S n and in this derivation the whale becomes one-dimensional (l w ). Gallos
and Argyrakis (2001) define the average “survival” probability (i.e., no encounter) as
Ps = e −Sn , where  = −log e (1−c v ). Thus, the average probability that the vessel
will encounter the whale is, P m = 1- P s . We use an example vessel (l v = 125 m,
b v = 20 m) and example length (l w = 16.5 m) and swimming speed (v w = 1.5 ms−1 )

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0.8
0.6
0.4
0.2
0.0

Average probability of a vessel and whale encounter

1.0

VANDERLAAN AND TAGGART: VESSEL COLLISIONS WITH WHALES

0

2

4

6

8

10

12

14

16

18

20

22

24

Vessel speed (knots)

Figure 4. The probability of a vessel and whale encounter, as a function of vessel speed,
within a 1 km2 domain estimated using a random walk model in two dimensions of a 16.5 m
whale swimming at 1.5 ms−1 in the presence of an example vessel (125 m length and 20 m
beam). The lines represent the domain with one whale and one vessel (solid), two vessels
(dash), and five vessels (dash dot).

of a whale within a 1 × 1 km domain. Vessel number and vessel speed (in this example vessels have identical dimensions and speed) are varied in the presence of one
whale in the domain. Although slow-moving vessels spend more time within the
domain than fast-moving vessels, this simple model (Fig. 4) demonstrates that the
encounter probability increases slowly as speed decreases from 24 knots or greater
and then begins to increase more rapidly as vessel speed continues to decrease toward
zero. This model represents an approximation of average encounter probabilities as
a function of vessel speed, yet it serves to illustrate that the encounter probability
does not increase with decreasing speed as simply as one might expect. Determining
such probabilities will be much more complex as the size and shape of the domain
(habitat) changes, as the number, sizes, and speeds of vessels and how they transit
the domain changes, as well as how the number, sizes, and speeds of whales and how
they move in the habitat changes.

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Slow-moving vessels may provide opportunity for whales to avoid a collision or
for vessel operators to avoid the whales. However, we are unaware of any compelling
evidence for either. According to Nowacek et al. (2004), NA right whales show
neither a behavioral response to the sounds of an approaching vessel nor to actual
vessels and suggest that NA right whales may be habituated to vessels noise and ignore
it. Southern right whales do not elicit “strong boat-avoidance” behavior (Best et al.
2001). Terhune and Verboom (1999) report an adult NA right whale turning into the
path of a small motor-vessel and cite Mayo and Marx (1990; though we cannot verify)
that on 64 of 138 occasions, NA right whales turned toward a parallel-running small
motorized vessel. For a vessel operator to avoid a collision with a whale, the whale
must first be detected and the operator must then maneuver to avoid the collision.
Large vessels navigating at low speed may not be able to maneuver successfully where
success is partially dependent on the operator’s ability to predict the movement of
the whale once detected. Whale detection is dependent on the surface profile of the
whale (right whales have no dorsal fin and thus minimum profile), unpredictable
whale behavior, lighting, meteorological conditions (day or night, fog, sea-state,
etc.), and observer bias (Hain 1997). Laist and Shaw (2006) report that small vessel
operators are unable to consistently detect and avoid manatees, and Best et al. (2001)
report a vessel collision with two or more whales where no avoidance action was taken
because the vessel operator anticipated the whales would dive to avoid the vessel.
We cannot dismiss vessel or whale avoidance of a pending collision as explaining
the few low-speed collision reports in the data we analyzed. We can suggest that the
paucity of low-speed collision reports is related to a paucity of vessels operating at low
speed. Our analysis of at-sea vessel speeds, associated with 1989–2002 mandatory
(>500 gross registered tons) and voluntary vessel reporting in the NW Atlantic,
shows 11.5% of the vessels navigating at ≤9 knots and 6.2% at ≤7 knots (n =
98,562; Eastern Canada Traffic Regulating System, ECAREG, unpublished data). It
is also possible that the few reports of vessel collisions with whales prior to 1960 (19
of the 294 records) may be related to (1) lower vessel speeds in earlier decades and
associated whale or vessel avoidance, and/or (2) collisions not being reported because
of an absence of interest in reporting and/or concern regarding vessel strikes. In the
first case, we have little quantitative evidence with which to reject the possibility,
although we note that of the nineteen pre-1960 collision reports, only six include
a vessel speed at the time of collision, and all six were ≥13 knots. Thirteen knots
is the contemporary mean vessel speed for the ECAREG data analysis noted earlier,
and it is consistent with the 14–16 knot contemporary average speed estimates of
Ward-Geiger et al. (2005). In the second case, we simply have no evidence to reject,
or not, the possibility.
In summary and acknowledging the uncertainties, our analyses provide compelling
evidence that as vessel speed falls below 15 knots, there is a substantial decrease in
the probability that a vessel strike to a large whale will prove lethal. The estimates
we provide can be used to consider the efficacy of vessel speed limits that have been
proposed in the United States (Federal Register (USA) 2006a) and are being proposed
elsewhere (United Nations Environmental Programme 2005, International Whaling
Commission 2006, Panigada et al. 2006).
ACKNOWLEDGMENTS
We are grateful to D. Kelley, C.C. Smith, B. Smith, and D. Gillespie for considerable
analytical insight and to J. Firestone, J. Corbett, M.W. Brown, A.R. Knowlton, J. Mullarney,

VANDERLAAN AND TAGGART: VESSEL COLLISIONS WITH WHALES

153

two anonymous referees, and D.A. Pabst for their critical appraisals. The data compilations of
D.W. Laist, A.R. Knowlton, J.G. Mead, A.S. Collet, M. Podesta, and A.S. Jensen and G.K.
Silber made the analyses possible. A. Serdynska and N. Helcl helped greatly. Funding for this
and related studies were provided by Canada NSERC, WWF-ESRF and HSP, and by U.S.
NOAA-NMFS.

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Received: 11 January 2006
Accepted: 22 August 2006

APPENDIX: ONE-DIMENSIONAL COLLISIONS WITHIN THE LIMITS OF THE ELASTIC
AND INELASTIC EXTREMES (SEE ONLINE SUPPLEMENTARY MATERIAL FOR
GREATER DETAIL)
Nomenclature
In all equations below, subscript 1 refers the vessel and subscript 2 refers to the
whale. The prime indicates the respective postcollision momenta and velocities. The
delta () indicates the change in either momentum (p) or time (t), and boldface
indicates vector quantities.
Newton’s Second Law is
F=

dp
,
dt

(1)

where F is force, p = mv is the momentum; the product of mass (m) and velocity (v).
Conservation of Linear Momentum: When no net external force acts on a system,
the total linear momentum of the system cannot change, thus,
m 1 v1 + m 2 v2 = m 1 v1 + m 2 v2 .

(2)

One-dimensional elastic collision:
An elastic collision is one where the postcollision kinetic energy of the system is
equal to the precollision kinetic energy of the system
1
1
1
1
m 1 v 21 + m 2 v 22 = m 1 v12 + m 2 v22 ,
2
2
2
2

(3)

which with Eq. 2, yields
v  2 − v  1 = − (v2 − v1 ) .

(4)

Hence, for elastic collisions the relative speed of recession postcollision equals the
relative speed of approach precollision.
Using Eq. 2 and Eq. 4, the postcollision velocity of the whale is solved as:
v2 =

2m 1 v1 + m 2 v2 − m 1 v2
.
m1 + m2

(5)

156

MARINE MAMMAL SCIENCE, VOL. 23, NO. 1, 2007

Substituting Eq. 5 into the momentum term of Eq. 1 yields


2m 1 v1 + m 2 v2 − m 1 v2
− m 2 v2
m2
d p  p p 2 − p 2
m1 + m2
=
˙
=
˙
=
˙
.
F =
d t t
t
t

(6)

One-dimensional inelastic collision: A perfectly inelastic collision is one where
only the momentum of the system is conserved and the postcollision velocities of the
two colliding bodies are equal and move as one body at velocity v  (i.e., v  = v  1 =
v  2 ). By using Eq. 2, the postcollision velocity is defined as:
v =

m 1 v1 + m 2 v2
.
(m 1 + m 2 )

(7)

Substituting Eq. 7 into the momentum term in Eq. 1 yields


m 1 v1 + m 2 v2
− m 2 v2
m2
d p  p p 2 − p 2
m1 + m2
=
˙
=
˙
=
˙
.
F =
d t t
t
t

(8)

Assumptions for the one-dimensional limiting cases first approximations: For both
types of collisions above, elastic and perfectly inelastic, we can reasonably assume
that both the mass and velocity of a large whale are much less than for a vessel; that
is, m 1
m 2 and v 1
v 2 . With these assumptions, the force equations (Eq. 6 and
Eq. 8) above simplify to
the elastic extreme
F ≈

v2
2m 2
1
v1 i f
t
v1

and

m2
 1,
m1

(9)

and

m2
 1.
m1

(10)

and the perfectly inelastic extreme
F ≈

v2
m2
v1 i f
1
t
v1

Thus, in either case, the forces involved in the collision are the product of the mass
of the whale and the speed of the vessel.
SUPPLEMENTARY MATERIAL
The following supplementary material is available for this article online:
One-dimensional collisions within the limits of the elastic and inelastic extremes.