THE MOMENTUM EQUATION AND A HEAD-¬–ON COLLISION

CAPT Joseph A. Derie, NAMS-­–CMS; AMS, SAMS; CMI
Co-­–Chair, NAMS Fishing Vessel Technical Committee
Southwest Passage Marine Surveys, LLC

I recently investigated a case where a PWC and a 19.5’ runabout collided in a head-­–on collision. The PWC had made a U-­–turn and came directly back at the runabout that was about 100’astern at the time of the turn. The PWC made no effort to turn and the operator of the runabout, faced with a difficult situation, cut the engine and coasted into the PWC rather than turning.

The accident resulted in the following damage to the PWC: total loss of the rub rail strip on the bow; a gouge to the hull-­–deck joint at the bow where the rub rail strip was torn off; and an approximately 18” scratch and gouge running up the sloping front bow area of the PWC. The PWC operator sustained a broken left leg, a broken left arm, a broken jaw and a concussion.

The runabout’s stem was undamaged so the gouge in the PWC that ran up the front of the vessel could only have been caused by the runabout’s bow eye. This was further indicated by some black marks from the PWC’s rub rail at the level of and behind the bow eye on the runabout. There were no injuries to the occupants of the runabout.

The physical evidence described above indicates the PWC was on plane when it hit the runabout and that the runabout was not on plane. The operator of the runabout stated that when he cut the engine he was going between 15-­–20 mph and that he was not planing because his wife doesn’t like it when the vessel is on plane.

His wife and the other occupant confirmed the vessel was not on plane just before the operator cut his engine. The damage to the PWC as well as the injuries to the PWC operator also appear to confirm this.

The crux of the opposing case was that the PWC was stopped or going at a minimum speed when the runabout collided with the vessel and the operator of the runabout could have and should have avoided the PWC. The plaintiff contended that the runabout’s operator’s failure to avoid the PWC was gross negligence, a violation of the USCG Inland Rules, Rule 5, Look Out; Rule 6, Safe Speed; and Rule 8, Action to Avoid Collision. In addition, he claimed that the physical evidence did not show that the PWC was on plane but did show that the runabout was definitely on plane.

Witnesses to the accident stated that the two vessels collided and basically stopped rather than one vessel than running over the other, or the vessels rebounding off each other. The damage to the PWC appears to confirm this. The accident investigation by the responding officers was minimal and a needle slap investigation of both vessels speedometers was not conducted.

Since the initial investigation had done nothing to establish the speed of either vessel at the time of impact and my investigation was not undertaken until several years later, there appeared to be no way to establish speed at this time.

However, given that both vessels stopped as a result of the impact and the PWC was on plane at the time of impact led me to consider the momentum equation as a way to determine the speed of both vessels.

The momentum equation is elementary physics. It says that the momentum of an object is equal to its mass times its velocity. If two vessels collide at different momentums (as is generally the case) the vessels will rebound or move accordingly. If the momentum of both vessels is equal in a head-­–on collision, there should be little or no movement after the collision (with some allowance for outside influences such as wind and water and both vessel center of gravity and center of buoyancy).

Using the momentum equation, I attempted to come up with a speed of the runabout at the time of the collision. Since the PWC was on plane at the time of the accident and I used 30 mph as its velocity. Below are my calculations.

1. Momentum = mass x velocity (mv). It should be noted that the mass of an object is its weight divided by the force of gravity. Since the weights of both vessels would have had to have been divided by the force of gravity, mathematically it didn’t matter whether I divided the weights or not. For simplicity I have chosen not to divide the weights of both vessels in the calculations.

2. The PWC is vessel 1. Its mass and velocity are m1 and v1. The weight of this vessel per the NADA guide is 474 lb. The operator weighed 150 lbs.
Assume 5 gallons of gas at 6 lb./gal = 30 lb.
Assume 16 lb. for the fire extinguisher and any other items on the vessel. Total weight = 670 lbs.

3. The runabout is vessel 2. Its mass and velocity are m2 and v2. The weight of this vessel per the NADA guide is 1690 lbs.
The assumed weight of the operator and two passengers is 680 lbs. Assume 10 gallons of gas at 6 lb./gal = 60 lb.
Assume 75 lb. for the fire extinguisher, PFDs and any other items on the vessel.

Total weight = 2265 lb.

4. If the PWC was going 30 mph, my assumed planing speed, then: m1v1 = m2v2
30 x 670 = 2265v2 v2 = 20100/2265
v2 = 8.9 mph

If the PWC was going 30 mph the runabout would have been going 8.9 mph in order to cause both vessels to stop in a head-­–on collision. Given the distance the vessels were apart when the PWC made its unsafe turn and came right back at the runabout, and the time for the runabout operator to understand the situation and react, this reduction in speed would appear to be consistent with the vessel going 15-­–20 mph before the operator cut his engine.

5. If the PWC was going 20 mph then:
m1v1 = m2v2
20 x 670 = 2265v2
v2 = 13400/2265
v2 = 5.9 mph

With the PWC going 20 mph the runabout would have been going 5.9 mph in order to cause both vessels to stop in a head-­–on collision.

6. If the PWC was going 40 mph, a possible planing speed for the PWC, then:

m1v1 = m2v2
40 x 670 = 26800v2 v2 = 26800/2265
v2 = 11.8 mph

This would also appear to be a possible speed for the runabout but leaves the possibility open that the vessel was going faster than the 15-­–20 mph stated. However, the injuries to the PWC operator would, I feel, argue for the slower speed of the calculation in paragraph 4 or 5 above.

7. What if the runabout was traveling 20 mph and didn’t slow down? Then:

m1v1 = m2v2
670v1 = 2265 x 20 v1 = 45300/670
v1 = 67.8 mph

If the runabout had been going 20 mph, the PWC would have had to have been going
67.8 mph to cause both vessels to stop in a head-­–on collision. This speed would not be consistent with the damage to both vessels and the injuries to the PWC operator only.

8. The above calculations are based on estimates of the total weight of both vessels at the time of the accident. To see how much the estimates of the occupants and other items of the vessels would have on the final calculations, the calculations in paragraph 2 above were redone with the PWC going 30 mph and the weight used for both vessels only that given in the NADA guide.

m1v1 = m2v2
30 x 474 = 1690v2
v2 = 14220/1690
v2 = 8.4 mph

If the PWC had been going 30 mph and the vessels had not been loaded, the runabout would have been going 8.4 mph at the time of the impact. This is versus the calculated 8.9 mph at the assumed loading with passengers, gas, etc. This also shows that in this case the loading of the vessels played a minor role in the momentum at the time of impact; the key figure is the weight of each vessel.

The above calculations combined with:

1) the damage to the PWC and the injuries to the PWC operator; as well as

2) the lack of damage to the runabout and lack of injuries to its passengers, appears to confirm that the runabout was going 15-­–20 mph before it stopped its engine just before the collision.

The case illustrates that in the limited circumstances of a head-on collision where both vessels are reported to have stopped after the collision, and the damage to both vessels and their occupants appears consistent with that, the momentum equation could be used to determine approximate vessel speeds. I used these calculations in court and I think I got my point across to the jury.

Courtesy NAMS Global enews