Skimming pebbles on water. But how does it happen?


Intro: Neither medium seems especially elastic in nature, so how do the stones bounce on water?


For the best results, a circular, flattish stone is best. It must be thrown so it is almost horizontal to the water’s surface, and also such that its trailing edge hits the water first. It is vital the action of the throwing imparts spin to the stone.

Any solid body moving through a liquid experiences forces that oppose its motion: these forces are proportional to the cross-sectional area of the body and the square of its speed. Although only a part of a skipping stone is actually moving through the water, with the rest travelling through air, these forces still have an effect on its forward motion.

There is a force exerted by the water at right angles to the spinning stone’s surface. It acts at the trailing edge of the spinning stone – because this is where the impact begins – and tends to turn the stone towards the horizontal. Because of its spin, though, the stone behaves like a gyroscope and refuses to change its orientation. Nevertheless, this force reduces the stone’s forward velocity somewhat.

There is also a force exerted by the water parallel to the stone’s surface, but this force is much smaller and so the stone’s velocity is barely changed by it on impact with the water. The net effect of these forces is that the stone flies from the water in a parabolic arc until it hits the water again and the whole process is repeated.

At each impact, the stone loses some of its kinetic energy, which is dissipated in the ripples that are created in the water. And as its velocity is gradually reduced, the impacts become closer together until the energy dissipated is greater than that lost in the impact and the stone sinks.

The minimum initial speed required by a stone varies with its inclination to the horizontal. Experiments show that skimming will not occur if the angle at which the flat surface of the stone hits the water is more than 45 degrees to the horizontal. The slowest speed for skimming is about 2.5 metres per second, when the inclination is about 20 degrees.

The fact that the water itself is not elastic is immaterial, but it is important that it gives way to the stone on impact.

Stones can also be skimmed on wet sand, and even on cloth-covered boards. In such cases, however, there is little or no give in the surface and the frictional grip at impact is sufficient to change the direction of the stone’s motion and also cause the stone to overcome the gyroscopic effect.

Skimming stones on water is an age-old pastime. The gunners of naval sailing ships worked out that it could be used to increase the range of cannonballs. These could be made to skip along the surface of the sea and hole enemy vessels near their waterline. However, to do this the cannon itself had to be near the sea surface, which meant the firing vessel needed perilously low ports. Indeed, some ships capsized after taking on water through those open ports. With cannonballs, the spin required by a skimming stone was unnecessary as their spherical symmetry precluded any gyroscopic effects.

Barnes Wallis’s famous bouncing bomb, created for the 1943 Dambusters raid on Germany in the Second World War, worked on the same principle. He had to use cylindrical bombs, though, so he figured out he could ensure their stability by giving them spin about a horizontal axis at right angles to their direction of motion on the water.


In a paper published in the journal Nature, in 2004, the French physicist Lydéric Bocquet and his team of researchers revealed some of the secrets of successful stone skimming. They found that the optimum angle of attack is 20 degrees. So, even when the stone is thrown horizontally, the leading edge should be 20 degrees higher than the trailing edge. This maximises the number of jumps by limiting the contact time between the stone and the water, which is proportional to the energy dissipated.

The thrower also imparts spin to the pebble, providing a gyroscopic effect that stabilises its flight and preserves the original angle of attack when it bounces. In the absence of spin, the water would impart a torque (or turning force) on the stone and, because the trailing edge is the first to make contact with the water, this would tend to make it tumble.

The actual physics of stone skimming is not perfectly understood. However, the bounce could be understood as a result of the conservation of momentum and Newton’s third law: when the stone exerts a force on the water, the water exerts an equal and opposite force on the stone. This lifting force is proportional to the density of the water, the surface area that is wetted and the square of the forward speed of the stone. Also, the bow wave created ahead of the stone when it strikes the water might act like a water-ski jump – helping to launch the next hop. This minimises the contact time between the stone and the water, which in turn maximises the number of jumps.

Although ensuring the optimal angle of attack as the stone strikes the water, and imparting just enough spin to maintain stable flight are important, there are other factors. Selecting the correct size and shape of stone and having a fast throwing arm are examples.

Given that the urge to skim stones has been with us for thousands of years and the rules – getting the greatest distance or number of bounces – have remained unchanged since the ancient Greeks, perhaps this should become an Olympic sport. In the meantime, the current world record stands at 51 skips, set by Russell Byars in Pennsylvania on 19 July 2007.

Britain, Government, Politics, Sport

Team GB offers a golden lesson in how to beat the world


RIo Medal Table

Rio Medal Table

OVER THE PAST FORTNIGHT in Rio at the Olympic Games, Team GB exceeded the highest hopes, spreading euphoria and joy even among many who thought they were uninterested in sport.

Across a huge range and spectrum of disciplines, Team GB athletes have shown what this country can achieve at its best. It amassed a hoard of gold medals to outstrip even China with its population of 1.4billion.

Yet, the performances we have all witnessed have given us much more than an excuse to fly the flag or by raising a toast to Britain’s highest ranking in the medal tables for 108 years. Team GB has offered a daily lesson in human virtue, gruelling effort and its rewards.

See: A collection of moments from the 2016 Rio Olympic Games – Olympic Rio Gallery 2016

Interviewed after their victories, these supremely dedicated athletes, for the most part modest and unassuming, have attributed their success above all to sacrifice, determination and an unrelenting work ethic.

In the words of the immortal Mo Farah, winner of four Olympic golds: ‘If you dream of something, have ambitions and are willing to work hard, then you can get your dreams.’

These are true role models for Britain’s young, too often captivated by dreams of the effortless, vacuous celebrity of the tawdry stars of reality TV.

If our athletes’ performance spurs them to emulate the commitment of the likes of Andy Murray on the tennis court, cyclist Laura Trott, boxer Nicola Adams, taekwondo’s Jade Jones and gymnast Max Whitlock, what rich rewards this country would reap. The showjumping gold won by 58-year-old Nick Skelton and older Britons, too, also offers a lesson about the importance of refusing to give up.


BUT isn’t there also a lesson for politicians in our athletes’ phenomenal success?

As one victor after another has been quick to acknowledge, Team GB owes at least a measure of its triumph to the ruthlessly effective way in which the British Olympic Association and UK Sport channelled state aid and lottery money into the disciplines most likely to yield the richest crop of medals.

Wasn’t the Olympic investment strategy – focusing on fields in which Britain is strong, the competition vulnerable and the rewards enormous – a perfect model for government priorities after Brexit?

The fact is that in commerce, as in sport, Britain has huge strengths – as even the most ardent of Remainers are beginning to admit.

Indeed, US banking giant JP Morgan has become the latest promoter of Project Fear to reverse its prediction that the FTSE-100 would plummet after a Brexit vote. Now it tells investors that British shares are the safest bet in Europe.

Certainly, there are tough negotiations ahead – and the sooner they start, the sooner the cloud of uncertainty hanging over the economy will lift.

But with the right focus, there is every hope that wonderful opportunities will open up when we’re free from Brussels interference to develop industries and services of our own choosing.

In Rio, our athletes have shown what huge rewards can be achieved through self-belief and hard work, backed up by clear-thinking administrators with their minds fixed on results.

The ministers in charge of negotiating Brexit should learn from them. Instead of fighting turf wars among themselves, Boris Johnson, Liam Fox and David Davis need a clear strategy to help us take on the world and win.

Team GB has shown how it can be done. Now the politicians in Britain must get on with it.

China, Foreign Affairs, Philippines, United Nations

China refuses to accept findings impinging on its sovereignty


Intro: The United Nations Convention on the Law of the Sea has no jurisdiction on territorial sovereignty, which, in China’s view, makes the court’s award illegal and invalid

China South China Sea China Air Patrol

Two Chinese Su-30 fighter jets take off from an unspecified location to fly a patrol over the South China Sea.

On July 19, an article appeared on this site entitled, China: An international ruling over the South China Sea.

Unilaterally initiated by the Philippines, the Arbitral Tribunal for the South China Sea announced its decree in July. China immediately responded by rejecting the court’s findings and the narrative here relates to why China has done so.

Firstly, Beijing insists that the Tribunal abused its authority by meddling in territorial issues. The disputes between China and the Philippines are about territorial sovereignty. China has held historical rights over the islands for some 2000 years without any disputes until the 1970s when the Philippines started to occupy China’s islands following reported discoveries of oil and natural gas in the region. According to its own rules, The United Nations Convention on the Law of the Sea has no jurisdiction on territorial sovereignty, which, in China’s view, makes the court’s award illegal and invalid.

Secondly, China decries that the ruling violated China’s legal rights. Beijing says that in light of international law, any country has rights to not accept dispute settlement imposed upon it on issues concerning territorial disputes. What is more, in 2006 China made a declaration excluding from arbitration matters concerning maritime delimitation. Over 30 countries (including the UK) have also made similar moves. In doing so, the award violated China’s rights.

Thirdly, Beijing claims that the court’s decree has harmed the international practice of peaceful settlement of disputes. China says it adheres to a peaceful foreign policy, one which seeks to settle disputes through negotiation and consultations. China has signed boundary treaties with 12 of its 14 land neighbours through bilateral negotiations in a spirit of equality and understanding. China has also been at pains to point out that it has reached consensus with the Philippines on settling their regional disputes through negotiation. However, the Tribunal turned a blind eye to it, damaging China’s goodwill.

And fourthly, China argues that the arbitration has intensified tensions in the region. Despite the disputes, the region remains peaceful with freedom of navigation unaffected. Beijing insists that the arbitration’s ruling will now accelerate tensions with countries outside the region and will be using it as an excuse for further interference and muddying the waters for their own interests.

European Union, NATO, Russia, Turkey, United States

Russia and Turkey’s rapprochement


Intro: For Russia, this is an opportunity to drive a hard wedge between Turkey, NATO and the EU

The unfolding diplomatic rapprochement between Russia and Turkey is likely to become a significant challenge for the European Union and NATO. For centuries now, these two countries have remained implacably opposed to each other. Efforts just a decade ago to forge a strategic partnership were curtailed by the civil war that has been raging in Syria. With Moscow clearly propping up Bashar al-Assad, Ankara either stayed out or implicitly supported his enemies. In more recent times, relations hit another low point last November when Turkish fighter jets shot down a Russian Su-24 bomber near the Syrian border for violating Turkey’s airspace. Russia imposed sanctions and the damage to relations between the two countries seemed irreparable.

But even before events last month in which an attempted military coup failed, President Erdogan had decided he could no longer afford a cold war of attrition and stalemate with Moscow and began making overtures with the Kremlin. The putsch appears to have expedited matters: yesterday Mr Erdogan met with Vladimir Putin to agree the normalising of relations. This will send shock waves through the EU at a time of unprecedented uncertainty.

For Russia, this is an opportunity to drive a hard wedge between Turkey, NATO and the EU and will help to abate Russian anger over the jet incident. President Putin must recognise in Mr Erdogan a leader cut from the same cloth – a democratically elected nationalist who has been behaving more like a despot.

Mr Erdogan’s ruthless purge of opponents after the thwarted coup has alarmed EU leaders who had encouraged Ankara to believe it could join the European Union at some future point and had pledged to introduce visa-free access for Turkish travellers to the Schengen area. No date, however, has ever been set or given for either and several EU countries have made it abundantly clear they would veto Turkey’s accession citing its human rights record, loss of press freedom and other economic shortcomings. Angela Merkel of Germany has been desperate to keep both options open in order to stop Turkey reneging on a deal to keep Syrian refugees from crossing into Europe.

But Mr Erdogan seems to have been cooling towards Europe, none of whose leaders have been to Ankara since the failed coup. Turkey’s leader is seeking alliances elsewhere. Improving relations between Russia and Turkey will have significant implications both for policy on Syria and for NATO itself. The US nuclear base at Incirlik is a key part of western defences, but, if Turkey were to leave its loss would be a serious blow to the organisation.

These developments will be concerning for European leaders. But for the Russian president this is a chance to cause fresh consternation in the capitals of Europe and in Washington. Mr Putin seems certain to grab a gift horse that couldn’t have come at a better time for his own interests.

Russia, Syria, United Nations, United States

Aleppo is on the brink of annihilation…


Intro: The Syrian government has demonstrated time and again how little it cares for international humanitarian laws

Aleppo is now more at a critical juncture that it has ever been since the start of Syria’s internecine civil war five years ago. Aleppo’s worsening situation comes at a bad moment when western attention has been turned sharply on terrorism in Europe and the impending US presidential election. Syria, however, is now demanding immediate attention too. What has been happening recently in Aleppo could be a decisive turning point in the conflict; any diplomatic hopes of whatever remain in negotiating a solution is fast deteriorating. An estimated 200,000 to 300,000 people are trapped in Aleppo’s eastern neighbourhoods, which are now entirely surrounded by Syrian government forces of Bashar al-Assad.  These troops are being assisted in their offensive by Russian air power and Iranian-controlled militias. Alarmingly, no food, no medical aid, nor any humanitarian assistance has been able to reach the population of Aleppo’s rebel-held territory for several weeks, because of the magnitude and intensity of the ongoing military onslaught.

Aleppo is of historical significance. It was once Syria’s second largest city, and it has become one of the key symbols of rebel resistance to the Assad regime since 2012. It has been a long-held objective of government forces to crush and obliterate Aleppo, and, if nothing is done to stop Assad’s forces advancing such a disaster seems imminent. That would not just be a defeat for the rebels, many of whom which have received western support, but perhaps an irreversible defeat for the uprising. Aleppo is staring into the abyss with the prospect of a new, humanitarian catastrophe of unprecedented proportions unfurling in Syria.


Map depicting Aleppo in Syria and surrounding countries.

With Aleppo encircled, the tragedy is being exasperated following the tightening of the knot in recent days by government forces whose aim has been to starve or empty it. Aleppo has been so ruthlessly shelled and bombed that it has become an inferno for those desperate people struggling among the ruins. There are hardly any doctors left in the city, and the last remaining hospital has been destroyed. UN agencies say food stocks are barely sufficient to last for more than three weeks.

The Syrian government and its Russian allies have resorted to a tactic of siege and starvation that has been used previously in Syria, but they are now doing it on a much larger and openly deliberate and provocative way. Their announcement of “humanitarian corridors” for civilians and rebels who would want to flee the area must be exposed as a cynical ruse. No one should be surprised that Aleppo’s population has not rushed towards these exit corridors, which have not in any case materialised on the ground. The Syrian government has demonstrated time and again how little it cares for international humanitarian laws. Assad’s machine of repression makes no distinction whatsoever between armed combatants and civilians. Tens of thousands of civilians have died while being held in detention centres. The announcement by Syrian and Russian officials without consulting or even warning UN agencies in advance is implicit proof that they want no external witnesses to their misdeeds.

Aleppo is on the brink of annihilation and the siege must be urgently lifted. International pressure is void of any credibility and its responses to a dire and stricken situation has been pitifully pathetic. It must put proper pressure on Russia to force Syrian troops to retreat, so that lives can be saved. The fate of Aleppo’s inhabitants, however, may now depend to a large degree on how global public opinion can now be mobilised. Saving Aleppo from utter destruction is not only a humanitarian imperative, but also central to any thin chance of a settlement in Syria ever being salvaged.

Arts, Drama, Puzzle

Lateral Thinking Drama: ‘Going For Gold’



A Lateral Thinking Drama & Puzzle

CARLA MET RODNEY while she was doing a postgraduate degree at Cornell University. She was studying the formation of atolls around the Pacific Rim, and he was doing research for a doctorate in geology. They had met at a Christmas party that her professor was giving at the end of term. Carla had been bored and was about to leave when she saw Rodney walk in. He had a confident air about him and she felt a strong, overpowering attraction. She threw her coat across the arm of a sofa and went to get a beer out of the fridge.

Rodney was very friendly when she introduced herself and soon they were talking in an excited manner about their respective studies and degrees. Carla told Rodney about her interest in the circular reefs formed by sunken volcanoes, and Rodney told her about his passion for what he called “the cooling rock”. He told her about a grant he was applying for which would allow him to complete his thesis in Tahiti.

“Tahit!” Carla exclaimed. “It’s a dream of mine to go to Tahiti. It’s a hot spot for atolls because of all the volcanic activity on the islands.”

“If I get this grant,” Rodney said, fuelled by alcohol and inspired by a growing sense of curiosity about the woman he was speaking to, “I’ll take you with me.”

“Is that a promise?” Carla asked.

Rodney put a hand over his heart and raised two fingers. “Scout’s honour,” he said, then bent down and kissed her on the forehead.


EIGHT MONTHS LATER, Carla and Rodney were married; and in the Autumn of that year, they moved to Tahiti.

Carla and Rodney rented a small, secluded bungalow tucked away in the jungle at the foot of an extinct volcano. They would set out in the morning to climb to the top and gather lava samples, then Carla would lay out a lunch of sandwiches and fruit salad. On one of their excursions, they discovered a deep gorge with a river flowing through it not far from the house. They came across an old tattered rope bridge suspended eighty feet above the raging water.

“This must be the bridge the old man was telling me about,” Rodney said.

“Oh, let’s cross it,” Carla pleaded.

“I can’t,” Rodney said. “I’m too heavy. Apparently the bridge can only hold a maximum of 125 lbs.”

“I don’t weigh that much,” Carla said and stepped out onto the bridge.

“Please come back, Carla,” Rodney pleaded. “You’re making me nervous.”


ONE DAY, a parcel arrived in the mail. It contained a letter from a solicitor informing Rodney that his father was dead. Carla was shocked.

“You told me your father died eight years ago,” Carla said to Rodney.

“In a manner of speaking, he did,” Rodney replied. “We fell out over money, which was typical because my father is a miser.”

“Was a miser,” Carla corrected him.

“We could have been very rich,” Rodney said.

“What do you mean?”

“My parents had a lot of money,” he explained. “When my mother died, I became the sole inheritor of my father’s wealth. I was nineteen and in my first year at college. I was already studying geology. My father wanted me to give up my studies and work as an apprentice is his textiles factory. He wanted to pass on the business, but I refused. I had no interest in textiles, or working for my father for that matter. Things escalated, and we came to blows. That was the last time I saw him. The papers came later, but he legally disowned me.”

“I can’t believe you never mentioned this to me,” Carla said and sat down. “Is that all the letter says?”

“No,” Rodney said. “There’s a receipt from the bank and a key. Something’s been transferred to a safety deposit box here, but it doesn’t say what it is.”


THAT AFTERNOON, Carla and Rodney went down to the bank and handed the slip to a teller. She escorted them into a safe and withdrew a long shallow box and handed it to Rodney. It was very heavy. Rodney looked at the bank teller and she excused herself so they could be in private. Rodney placed the box on a shelf and inserted the key. He opened the lid and his eyes grew wide in amazement.

“What is it?” Carla asked.

Rodney withdrew two shiny gold ingots from the box and held them up for her to see. He turned them over in his hands. On the bottom was etched the weight of each bar of gold. They weighed nine pounds each.

Carla put the ingots into her bag. They thanked the bank teller and drove to their favourite restaurant. Rodney ordered a bottle of champagne and placed the two ingots on the table where he could get a better look at them. When the waiter glanced at the gleaming bars on the table, Rodney grabbed him by the arm and said, “God, man. It’s pure gold.”

The waiter was stunned. He asked if he could hold one, then he called the bartender over. Soon everybody at the restaurant had got up to take a look. Carla had never seen so many greedy eyes. She started to feel penned in by the crowd, unable to breathe. She wriggled out of her seat and pushed her way out of the circle of people that had gathered at their table. She went to the bathroom and splashed cold water on her face. This was bad news; she wished the letter had never arrived.

That night, Rodney and Carla had a fight. Rodney didn’t feel like sleeping so he took the car and drove down to the ocean to walk along the beach. He was ashamed of his reaction to the gold, and the sea had always had a calming effect on him. He promised to take the ingots back to the bank the very next morning and sell them. He decided to donate half the money to an environmental charity, and to put the rest into a separate account for when he and Carla had children.

Back at the bungalow Carla paced the porch waiting for Rodney to return. It had started to rain and Carla held her hand out to catch the cool drops in her palm. She felt nervous being left alone with the gold ingots and waited impatiently for her husband to come home.

While Carla was out on the front porch, two men broke into the house. They had heard about the gold from a friend who had been at the restaurant earlier that evening. They had been watching the house and seen the car speed down the driveway and figured that nobody was home. Carla jumped when she heard their voices, then crouched down under the living room window. She heard a man say, “Where do you think they put it?”

“How am I supposed to know?” another man answered.

“Try the bathroom,” the first man said. “I’ll check the bedroom.”

Carla wanted to jump off the porch and run away, but then she remembered what Rodney had said earlier that evening. For the last year Carla had been wanting to get pregnant, but Rodney always said they couldn’t afford a baby. That night, he had agreed that now was a good time. Carla knew that the only reason he had said this was because of the gold, and she had become mad at him for being so materialistic. But now she understood why it was so important to him; the money would allow them to start a family. Suddenly the gold became very important to her too.

Carla crept back inside the bungalow and down the hall towards the kitchen. She had put the gold ingots back inside her bag after Rodney had stormed out, and left it on the kitchen table. Carla seized her bag and ran back down the hall towards the front door. As she passed the doorway to her bedroom, one of the robbers looked up and saw her.

“It’s her!” he yelled. “Don’t let her get away!”

Carla burst through the front door and out into the dark night. She could hear the robbers close on her heels, yelling and stumbling through the undergrowth. She found herself running down the path towards the gorge and could hear the flowing river in the distance ahead of her. She needed a way to elude the robbers and thought of the old rope bridge that crossed the chasm. She remembered that the bridge could only withstand a maximum of 125 lbs. She had weighed herself recently and knew that she weighed 110 lbs.

When Carla got to the bridge, she stopped. The robbers were quickly gaining on her. She took her shoes off, then rummaged through her bag and withdrew the two gold ingots. She knew they weighed 9 lbs each. She didn’t want to part with them but she couldn’t carry both bars across at the same time or the bridge would break. She realised that she didn’t have enough time to make two trips, and she couldn’t throw the bars because the ravine was too wide. Carla held her breath and listened to the river roaring eighty feet below.

She looked back towards the jungle. She could just see the two men emerging from the trees. Suddenly she had an idea; it was easy after all. She was going to escape.

How did Carla manage to cross the gorge with both gold ingots?


© MD 2016: all rights reserved





Europe, France, Government, Islamic State, Society, Terrorism

Europe and Islamist attacks


Intro: President François Hollande of France may think that declaring war on the extremists will shore up his own fragile political position

THE INSTINCTIVE RESPONSE on horrors such as those that have taken place in France and Germany in recent days is to look for a pattern, a narrative that might go some way to explain the inexplicable.

The brutal and bloody murder of an 86-year-old priest in Normandy invites such thinking, since it follows years of attacks on Christians in the Middle East: first by al-Qaeda and then by Islamic State of Iraq and the Levant (ISIL). Is radical Islam seeking a war with Christianity?

The very suggestion or notion of such a conflict between faiths would delight followers of ISIL, but it is hard to reconcile with that group’s dreadful persecution of fellow Muslims. ISIL has killed many more Muslims than it has Christians or Jews.

Or are the Islamists targeting Western liberal values more broadly, seeking to reinstate the Islamic Caliphate that once existed across the Middle East and parts of Southern Europe?

If so, that end has been poorly served by the enormity and mayhem in Normandy and Bavaria, lands that were never home to Muslims in the middle ages and which have only come to have Muslim residents as a result of those liberal Western values.

Seeking some kind of explanation for the evil that has been perpetrated is perfectly natural, but we should not impute too much calculation or design to those individuals who carry out such heinous crimes.

Whilst we may look for explanations the truth is there is no rationale or logic, nor any coherent argument in explaining away why Europe is suffering such appalling atrocities on its streets. These are the acts of inadequate and disturbed individuals with a nihilistic desire to destroy anything that challenges them and their ill-formed and warped idea of the world.

We must harden our defences against such acts, but we should be wary of the idea that those acts represent a clash of cultures – for that suggests some sort of parity between irrational extremist ideology on the one hand and a civilisation of shared traditions developed over thousands of years on the other.

President François Hollande of France may think that declaring war on the extremists will shore up his own fragile political position. Such a response, however, also risks validating the arguments of Marine Le Pen’s National Front (i.e. that the French establishment has failed to face up to the existential threat of terrorism).

Security and intelligence operations should be reviewed in the face of these latest attacks, particularly as the numerous intelligence agencies that operate in France are highly dysfunctional and disjointed. Great care must be taken not to dignify the attackers or their pathetic dreams of grandeur. They are murderers only deserving of contempt.