Speed of a Spear: Troy

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Abstract

The purpose is to determine the speed of a spear thrown by the mighty Achilles at Captain Tecton in the movie Troy (2004).

Introduction

The movie surrounds the events of the Trojan War as described in Greek myths. In Troy, the Greeks fight the Trojans because Helen, the Queens of Sparta, leaves her husband Menelaus to stay with the Trojan prince. When Menelaus finds out about his wife, he sends his brother and an army of 50,000 soldiers to Troy. The Greek army is very formidable with the help of Achilles; however, they eventually fall to Hector, the prince of Troy.

In Troy, Achilles, the hero of the film, possesses superhuman powers. Achilles displays his immense strength in one scene by throwing a spear over an extremely long range directly into the throat of an oncoming Trojan soldier, Captain Tecton, riding on horseback.

To find the initial velocity and height, we are going to use calculations based on the equations of projectile motion. For the horizontal velocity, we will start with:

x = x₀ + v_0xt
v_x = v_0x
a_x = 0 m/s²

Since there is no acceleration (a) in the x direction, velocity (v) is constant and is equal to the initial velocity. The horizontal displacement (x) is estimated from Achilles's position to Captain Tecton's dead body after he falls off his horse. The time (t) measures how long it takes for the spear to travel from Achilles's hand to Captain Tecton's throat.

For the height above Captain Tecton from where Achilles throws the spear:

y = y₀ + v₀t – ½ gt²
a_y = – g

Gravity (g) is the only acceleration for the vertical component. The height (y) is measured from the Achilles to a certain distance below him that is level with Captain Tecton's dead body after he falls of his horse.

Screen Captures

Procedure

Play the scene using a media player such as VLC.
Load the video into a video editor such as VirtualDub.
Count the number of frames and seconds from when Achilles lets go of the spear to when Captain Tecton is impaled.
Approximate the horizontal displacement in the scene where it shows both Achilles and the dead Captain Tecton at the same time.

Analysis

We played the scene in VirtualDub from when the spear leaves Achilles's hand and hits Captain Tecton. VirtualDub allowed to us count the the number of frames and seconds that scene took up after we edited it. VirtualDub displayed the scene to be 45 frames and 1.88 s.

It is assumed that Achilles throws at a 45 degree angle. We estimated that the total horizontal distance from Achilles and Tecton's corpse was equal to about 1.5 times the length of a long Manhattan block (approx. 250 meters). This would end up being 375 meters.

To find the horizontal velocity, use:

x = x₀ + v_0xt

Let x₀ equal zero and set the equation equal to v_0x

v_0x = x/t

Substitute values.

v_0x = (375 m)/(1.88 s)

Solve for v_0x.

v_0x = 199.47 m/s

This is the horizontal component of the spear's velocity. To find the velocity of the spear itself, we will use:

v = v₀cos(θ)

and assume a maximal launch angle of 45°.

(199.47 m/s) = v₀cos(45°)
v₀ = 282 m/s

Conclusion

The velocity of the spear was 282 m/s or 630 mph -- as fast as a jet airplane.

Sources of Error

The horizontal distance from Achilles to Captain Tecton was estimated.
Drag forces acting on the spear were not taken into account.

Janan Liu, Stephen Singh, and Andrew Zheng -- 2005

Physics on Film