Monday, February 22, 2010

Mighty David and giant Goliath--physics

Unlike some other events described in the Bible [a burning bush not consumed or the parting of the Red Sea[ there is some sound physics in the David and Goliath story.

Here's the account [1st Samuel Chapter 17] [New International Version]...

1 Now the Philistines gathered their forces for war and assembled at Socoh in Judah. They pitched camp at Ephes Dammim, between Socoh and Azekah.

2 Saul and the Israelites assembled and camped in the Valley of Elah and drew up their battle line to meet the Philistines.

3 The Philistines occupied one hill and the Israelites another, with the valley between them.

4 A champion named Goliath, who was from Gath, came out of the Philistine camp. He was over nine feet [a] tall.

5 He had a bronze helmet on his head and wore a coat of scale armor of bronze weighing five thousand shekels [b] ;

6 on his legs he wore bronze greaves, and a bronze javelin was slung on his back.

7 His spear shaft was like a weaver's rod, and its iron point weighed six hundred shekels. [c] His shield bearer went ahead of him.

8 Goliath stood and shouted to the ranks of Israel, "Why do you come out and line up for battle? Am I not a Philistine, and are you not the servants of Saul? Choose a man and have him come down to me.

9 If he is able to fight and kill me, we will become your subjects; but if I overcome him and kill him, you will become our subjects and serve us."

10 Then the Philistine said, "This day I defy the ranks of Israel! Give me a man and let us fight each other."

11 On hearing the Philistine's words, Saul and all the Israelites were dismayed and terrified.

12 Now David was the son of an Ephrathite named Jesse, who was from Bethlehem in Judah. Jesse had eight sons, and in Saul's time he was old and well advanced in years.

13 Jesse's three oldest sons had followed Saul to the war: The firstborn was Eliab; the second, Abinadab; and the third, Shammah.

14 David was the youngest. The three oldest followed Saul,

15 but David went back and forth from Saul to tend his father's sheep at Bethlehem.

16 For forty days the Philistine came forward every morning and evening and took his stand.

17 Now Jesse said to his son David, "Take this ephah [d] of roasted grain and these ten loaves of bread for your brothers and hurry to their camp.

18 Take along these ten cheeses to the commander of their unit. [e] See how your brothers are and bring back some assurance [f] from them.

19 They are with Saul and all the men of Israel in the Valley of Elah, fighting against the Philistines."

20 Early in the morning David left the flock with a shepherd, loaded up and set out, as Jesse had directed. He reached the camp as the army was going out to its battle positions, shouting the war cry.

21 Israel and the Philistines were drawing up their lines facing each other.

22 David left his things with the keeper of supplies, ran to the battle lines and greeted his brothers.

23 As he was talking with them, Goliath, the Philistine champion from Gath, stepped out from his lines and shouted his usual defiance, and David heard it.

24 When the Israelites saw the man, they all ran from him in great fear.

25 Now the Israelites had been saying, "Do you see how this man keeps coming out? He comes out to defy Israel. The king will give great wealth to the man who kills him. He will also give him his daughter in marriage and will exempt his father's family from taxes in Israel."

26 David asked the men standing near him, "What will be done for the man who kills this Philistine and removes this disgrace from Israel? Who is this uncircumcised Philistine that he should defy the armies of the living God?"

27 They repeated to him what they had been saying and told him, "This is what will be done for the man who kills him."

28 When Eliab, David's oldest brother, heard him speaking with the men, he burned with anger at him and asked, "Why have you come down here? And with whom did you leave those few sheep in the desert? I know how conceited you are and how wicked your heart is; you came down only to watch the battle."

29 "Now what have I done?" said David. "Can't I even speak?"

30 He then turned away to someone else and brought up the same matter, and the men answered him as before.

31 What David said was overheard and reported to Saul, and Saul sent for him.

32 David said to Saul, "Let no one lose heart on account of this Philistine; your servant will go and fight him."

33 Saul replied, "You are not able to go out against this Philistine and fight him; you are only a boy, and he has been a fighting man from his youth."

34 But David said to Saul, "Your servant has been keeping his father's sheep. When a lion or a bear came and carried off a sheep from the flock,

35 I went after it, struck it and rescued the sheep from its mouth. When it turned on me, I seized it by its hair, struck it and killed it.

36 Your servant has killed both the lion and the bear; this uncircumcised Philistine will be like one of them, because he has defied the armies of the living God.

37 The LORD who delivered me from the paw of the lion and the paw of the bear will deliver me from the hand of this Philistine." Saul said to David, "Go, and the LORD be with you."

38 Then Saul dressed David in his own tunic. He put a coat of armor on him and a bronze helmet on his head.

39 David fastened on his sword over the tunic and tried walking around, because he was not used to them. "I cannot go in these," he said to Saul, "because I am not used to them." So he took them off.

40 Then he took his staff in his hand, chose five smooth stones from the stream, put them in the pouch of his shepherd's bag and, with his sling in his hand, approached the Philistine.

41 Meanwhile, the Philistine, with his shield bearer in front of him, kept coming closer to David.

42 He looked David over and saw that he was only a boy, ruddy and handsome, and he despised him.

43 He said to David, "Am I a dog, that you come at me with sticks?" And the Philistine cursed David by his gods.

44 "Come here," he said, "and I'll give your flesh to the birds of the air and the beasts of the field!"

45 David said to the Philistine, "You come against me with sword and spear and javelin, but I come against you in the name of the LORD Almighty, the God of the armies of Israel, whom you have defied.

46 This day the LORD will hand you over to me, and I'll strike you down and cut off your head. Today I will give the carcasses of the Philistine army to the birds of the air and the beasts of the earth, and the whole world will know that there is a God in Israel.

47 All those gathered here will know that it is not by sword or spear that the LORD saves; for the battle is the LORD's, and he will give all of you into our hands."

48 As the Philistine moved closer to attack him, David ran quickly toward the battle line to meet him.

49 Reaching into his bag and taking out a stone, he slung it and struck the Philistine on the forehead. The stone sank into his forehead, and he fell facedown on the ground.

50 So David triumphed over the Philistine with a sling and a stone; without a sword in his hand he struck down the Philistine and killed him.

51 David ran and stood over him. He took hold of the Philistine's sword and drew it from the scabbard. After he killed him, he cut off his head with the sword. When the Philistines saw that their hero was dead, they turned and ran.

52 Then the men of Israel and Judah surged forward with a shout and pursued the Philistines to the entrance of Gath [g] and to the gates of Ekron. Their dead were strewn along the Shaaraim road to Gath and Ekron.

53 When the Israelites returned from chasing the Philistines, they plundered their camp.

54 David took the Philistine's head and brought it to Jerusalem, and he put the Philistine's weapons in his own tent.

55 As Saul watched David going out to meet the Philistine, he said to Abner, commander of the army, "Abner, whose son is that young man?" Abner replied, "As surely as you live, O king, I don't know."

56 The king said, "Find out whose son this young man is."

57 As soon as David returned from killing the Philistine, Abner took him and brought him before Saul, with David still holding the Philistine's head.

58 "Whose son are you, young man?" Saul asked him. David said, "I am the son of your servant Jesse of Bethlehem."


a. 1 Samuel 17:4 Hebrew was six cubits and a span (about 3 meters)
b. 1 Samuel 17:5 That is, about 125 pounds (about 57 kilograms)
c. 1 Samuel 17:7 That is, about 15 pounds (about 7 kilograms)
d. 1 Samuel 17:17 That is, probably about 3/5 bushel (about 22 liters)
e. 1 Samuel 17:18 Hebrew thousand
f.. 1 Samuel 17:18 Or some token ; or some pledge of spoils
g. 1 Samuel 17:52 Some Septuagint manuscripts; Hebrew a valley

Now, what are the physics?

Here is a decent description followed by a more complex explanation from Wolfram.

"David & Goliath… an Engineer’s Perspective"


Random Benjamin


Rainy Skys

Anyone who attended Sunday school as a child, or indeed I would contend the average American in general has heard the Biblical story of David and Goliath. This is an incredible story of a young Israelite shepherd boy full of God’s power defeating a giant Philistine, Goliath, in combat. Such a story is interesting on many points, but there is one in particular that few people talk about that I would like to analyze; the physics behind the event.

The full Biblical passage may be found in 1 Samuel 17; I highly recommend reading it in its entirety to get the complete context. But in the interest of conciseness and articulation I will give only a short summary so that we may quickly proceed to the physics.

The Philistines had gathered an army to attack Israel and their champion was a fearsome man named Goliath. The Israelite army had also mustered in defense and every day would approach the Philistines for battle, but instead of fighting Goliath would challenge them, “Do you need a whole army to settle this? Choose someone to fight for you, and I will represent the Philistines. We will settle this dispute in single combat! If your man is able to kill me, then we will be your slaves. But if I kill him, you will be our slaves!” The Bible goes on to say a few verses later, “When [King] Saul and the Israelites heard this, they were terrified and deeply shaken.”

So all the Israelite wimps sat around and did nothing; until one day along came this kid (David), bringing food and other supplies to his brothers in the army and he heard all the stuff Goliath was saying and was amazed at the pathetic showing of his country-men. He went to King Saul and said, “I’ll go fight this Philistine!” So he eventually got permission and went out to fight Goliath.

In analyzing this account, I would like to find out if modern physics supports the basic idea of how David defeated Goliath. This question is somewhat subjective because it is very hard to acquire any accurate data or measurements. Also the force required to bring down Goliath must be estimated rather arbitrarily since it is obviously not simply equal to his overall mass times acceleration.

The Bible says Goliath was, “a giant of a man, measuring over nine feet tall! He wore a bronze helmet and a coat of mail that weighed 125 pounds. He also wore bronze leggings, and he slung a bronze javelin over his back. The shaft of his spear was as heavy and thick as a weaver’s beam, tipped with an iron spearhead that weighed fifteen pounds.” I proceeded to research the Greek and Hebrew accounts. In Hebrew, Goliath is listed as 6 cubits and 1 span tall, which totals about 9.75 feet or 3 meters; the Greek version and Dead Sea Scrolls list him as 4 cubits and 1 span, which totals about 6.75 feet or 2 meters. The weight of his armor (125 pounds) is listed in Hebrew as 5,000 shekels which equals 57 kilograms. The weight of his spearhead (15 pounds) is listed in Hebrew as 600 shekels which equals 6.8 kilograms. I can also infer that a “giant of a man” standing approximately 7 to 9 feet tall would feasibly weigh 300 to 400 pounds (137 to 183 kilograms) – mostly in muscle mass. Goliath is a big man, it would take a lot to defeat him; it is more understandable from this point of view why the Israelites were so afraid of him.

Another verse which provides us with a lot of physics information on the event reads as follows. “As Goliath moved closer to attack, David quickly ran out to meet him. Reaching into his shepherd’s bag and taking out a stone, he hurled it from his sling and hit the Philistine in the forehead. The stone sank in, and Goliath stumbled and fell face downward to the ground. So David triumphed over the Philistine with only a stone and sling. And since he had no sword, he ran over and pulled Goliath’s sword from its sheath. David used it to kill the giant and cut off his head.”

So we know that David increased the velocity of the stone by adding to the initial velocity by running, we also see that the stone hit Goliath in the forehead, a relatively unprotected area as the helmets of the time did not cover much of the forehead. Goliath apparently had some forward momentum, enough that even after the stone had sank into his forehead and knocked him unconscious, his body still continued moving forward until crashing into the ground moments later. The stone must have done considerable damage but it did not kill Goliath, David ran over and picked up Goliath’s sword and killed him with it.

This is all based on the presumption that the concept of a sling actually works, a concept known in physics as centripetal force. As an object moves around in a circular motion it is constantly accelerating tangent to changing velocity with constant speed. The sling David used probably was a leather pouch attached at both ends with leather strings. After loading a stone into the pouch one could whirl the sling around your head and with precise timing release one of the strings – this would instantly release the centripetal force, sending the stone on a trajectory initially tangent to the orbit and continuing on a straight line towards Goliath. The velocity may be calculated by taking 2π times the radius (approximately 1 meter) divided by the period (time it takes to complete one rotation, approximately 1 second). This velocity may then be substituted into the equation for centripetal acceleration (ac = velocity2/radius) which then equals 39.5 meters per second per second.

Goliaths force (Fg) is approximated by his mass (average is 160 kg) times acceleration (maybe 2 meters per second per second); so Fg = 320. Where the force of the stone hitting Goliath can be found by multiplying the acceleration due to the centripetal force exerted by the sling (which I already found) multiplied by the weight of the stone (probably about 1 kg.); (Fs) = ac * m, (Fs) = 39.5. This around an eighth as much force as Goliath, but that is a lot of force and it was applied and transferred in an instant over a small area. Much like a karate expert breaking slabs of concrete – a large amount of force applied over a short amount of time in a small area will result in the most force being transferred and the most damage being inflicted. Remember all this damage is centered on a small, unprotected area of the forehead.

I can theorize the stone would have also had a lot of kinetic energy. Kinetic energy equals ½ times mass times velocity squared. A professional baseball pitcher can throw a ball close to 100 mph; a sling will increase the velocity exponentially. Even though David is considered to be fairly young at this point in his life, he would have been very familiar with a sling as his primary weapon and tool as a shepherd. This suggests the stone slung by David has a large amount of velocity – which is then squared when put in the formula. As for the other part of the formula, one half times a mass of approximately 2 kg. equals one, so the stone has a lot of kinetic energy.

From this information I can conclude that the stone had more than enough potential to do the damage necessary to bring down a man, even a man as big as Goliath. I would contend it would have been a large enough blow to kill most people; however, the stone doesn’t have to kill Goliath, the Bible says David killed Goliath by cutting of his head. Therefore, it is very reasonable to conclude that the physics behind this story support the validity of the account.

For a more complex explanation see...

Energy of a Slingshot: David and Goliath from Wolfram Demonstrations Project

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