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Opus in profectus

# Conservation of Momentum

## Practice

### practice problem 1

In the 1996 action-adventure movie Eraser, Arnold Schwarzenegger plays a US Marshall working for the Witness Relocation Program. He obtains a high-tech weapon that launches projectiles at "nearly the speed of light". Arnold wields the weapon like a shotgun, firing it several times from an unbraced, standing position. Bad guys are then seen flying through the air (for several meters) after being hit. Explain the physical impossibility of operating a weapon like this in the manner described above.

#### solution

Arnold Schwarzenegger does not appear to obey the law of conservation of momentum. If he uses weapons that fire projectiles forward, he and the weapons must travel backward, or recoil.

The total momentum of the projectiles and Arnold before either weapon is fired is zero since neither is really going anywhere. After a weapon is fired, the total momentum must still be zero since momentum is a conserved quantity. This only works out mathematically if we assign a positive value to the projectile's forward momentum and a negative value to Arnold's backward momentum. A positive momentum plus an equal negative momentum can add up to zero.

The phrase "nearly the speed of light" is ambiguous, so let's keep things simple and just go with the nearest order of magnitude — 108 m/s. (The speed of light in a vacuum is 3.00 × 108 m/s.) Arnold Schwarzenegger probably weighs more than 100 kg and with two heavy guns he certainly weighs more than 100 kg, but again let's keep things simple and go with a nice round number — 100 kg. How much do the projectiles weigh? The guns in this movie look something like heavy machine guns. These weapons use bullets weighing around 40–50 g, but again let's make life easy and just go with a simple number like 10 g. There's no reason to get real precise here. This weapon will prove to be ridiculous no matter what.

 mbullet = 10 g = 10−2 kg vbullet = 108 m/s marnold = 100 kg = 102 kg varnold = ?
 ∑pbefore = ∑pafter 0 = mbulletvbullet + marnoldvarnold −marnoldvarnold = +mbulletvbullet −(102 kg)varnold = +(10−2 kg)(108 m/s) varnold = −104 m/s

For comparison, the speed needed to escape the gravitational pull of the Earth is 11,000 m/s or about 104 m/s. Arnold would be on his way to the Moon after firing one of these weapons — assuming his arms stayed attached.

Arnold shoots two projectiles forward at "nearly the speed of light". His massive bodybuilder arms separate from his torso in a literal Farewell to Arms. (His guns fly away with the guns.) Each arm is tied to a weapon by a meaty Austrian trigger finger. (The wieners from Wien.) The arm-gun combos punch through the wall behind him in quick succession, tearing through the atmosphere faster than a rocket to the Moon. Shock waves shatter windows and frighten livestock across several states. Arnold's arms burn up in the atmosphere like reverse meteorites. The weapons continue on into space where they orbit the Sun for the rest of eternity. "Eraser is the wrong name for this movie," Arnold realizes. "We should have called it Total Recoil." Blood spews from his empty arm sockets. With his last remaining breaths he recites his most famous catch phrases. "It's not a toomah. I'll be back. Get to the choppah. You should not drink and bake. Put that cookie down."

### practice problem 2

If a significant portion of the population of China (say one billion people) simultaneously jumped up at 1 m/s, how fast would the Earth recoil? Assume an average mass of 60 kg per person. (Do not use a calculator to compute your answer. This is meant to be an order of magnitude calculation.)

#### solution

Before all these people jump, they are at rest with respect to the Earth. The Earth-people system has a total momentum of zero. After these billion people jump, the total momentum must still be zero. This is only possible if the momentum of the Earth is opposite and equal to the momentum of the people. This is an example of a recoil problem.

 +ppeople = −pearth +mpeoplevpeople = −mearthvearth

The mass of the Earth is 5.9736 × 1024 kg, a number that is stupidly precise for this silly question. Let's just call it 6 × 1024 kg and move on.

 +(60 × 109 kg)(1 m/s) = −(6 × 1024 kg)vearth vearth = −10−14 m/s

This speed is equivalent to moving the diameter of ten protons in one second. The Earth would move, but it would be practically impossible to detect this.

### practice problem 3

Write something different.