# Kinetic Energy, Bullets and Collisions: A Maths Tutorial

- define the kinetic energy of a mass \(\normalsize{m}\) moving with velocity \(\normalsize{v}\)
- learn about the Law of Conservation of Energy and its implications for bullets
- see how physicists analyse elastic particle collisions.

## Definition of kinetic energy

Physicists regard kinetic energy as energy due to motion, and sometimes also refer to the amount of work that a moving particle can do. For a single particle of mass \(\normalsize{m}\) moves with velocity \(\normalsize{v}\), its**kinetic energy**is officially defined as\[\Large{T=\frac{1}{2}mv^2}.\]The dependence of the kinetic energy \(\normalsize{T}\) on the velocity \(\normalsize{v}\) is quadratic. This is a marked difference from what we saw with the momentum \(\normalsize{p=mv}\), which depends linearly on velocity. So if we double the speed of a particle, its momentum doubles, but its kinetic energy is multiplied by four. If we triple the speed of a particle, its momentum triples, but its kinetic energy is multiplied by nine.In general energy may shift from one form to another during interactions, but the Law of Conservation of Energy states that the total amount of energy is conserved. With a swinging pendulum for example, the mass has at the top of its swing no kinetic energy, but only potential energy, while at the bottom of its swing it has the most kinetic energy and the least potential energy. In real life, most moving particles lose energy through friction, so that their kinetic energies eventually are transferred to heat energy.

## The difference between momentum and energy

Somewhat simplistically, the momentum of a moving particle is a measure of how much push it has; whereas the kinetic energy is a measure of how much damage it can do. This explains why martial artists practice to punch fast. With a small increase in the speed of a punch, the power to damage increases quadratically.This is also the reason why bullets are so deadly, even if they are relatively light; they have a high speed. Rifle bullets typically have two or three times the speed of pistol bullets, so that even with the same size, the energies of rifle bullets are between four and nine times the energy of pistol bullets, making them correspondingly more deadly.Here are some figures: the**caliber**of a bullet is usually measured in inches, unless explicitly stated in millimetres, and refers to the diameter of the circular cross-section.

Firearm | Caliber | Muzzle energy (joules) |
---|---|---|

air gun spring | .177 | 20 |

air gun PCP | .22 | 40+ |

pistol | .177 | 159 |

pistol .357 Magnum | .177 | 873 |

rifle | .30 | 2,000 |

Q1(E): The Lone Ranger’s rifle shoots bullets at 3000 ft/sec, while Butch Cavendish’s rifle shoots the same size bullet at 4000 ft/sec. How much more energy do Butch’s bullets have?Q2(M): If the Lone Ranger’s rifle shoots .30 caliber bullets at 3000 ft/sec, and his side-kick Tonto’s pistol shoots .20 caliber bullets at 1000 ft/sec, then how much more energy do the Lone Ranger’s bullets have? [Let’s assume that the bullets have the same lengths.]

*Gravity*, starring Sandra Bullock, where even light particles can do significant damage if they are moving swiftly enough, (even if this might stretch the bounds of what we can expect orbiting satellites to experience).

## What happens when things bounce, or go bang

## An explicit physics computation (advanced)

before collision | after collision | |
---|---|---|

total momentum | \(5 \times 3 + 0 \times 1 = 15\) | \(3 w_1 + w_2\) |

total kinetic energy | \(\frac{1}{2} \times 3 \times 5^2 + \frac{1}{2} \times 1 \times 0^2= \frac{75}{2}\) | \(\frac{1}{2} \left( 3 w_1^2 + w_2^2\right)\) |

Q3(M): What happens if we change \(\normalsize{m_2}\) to \(\normalsize{5}\)?

Q4(M): What happens if we change \(\normalsize{m_2}\) to \(\normalsize{8}\)?

## Answers

A1.The ratio is \(\normalsize{4000^2/3000^2}\), or \(\normalsize{16/9}\). Cavendish’s bullets have almost twice as much energy.A2.In this situation things are more complicated since not only are the velocities different, the masses are too. Let’s remember that the mass of a cylindrical bullet of a fixed length is proportional to the cross-sectional area, which is proportional to the square of the radius, and so also to the square of the diameter. Thus the ratio of the energies of the bullets is the ratio of \(\normalsize{\frac{1}{2}(.3)^2(3000)^2}\) to \(\normalsize{\frac{1}{2}(.2)^2(1000)^2}\). This is the ratio of \(\normalsize{81/4}\), or more than \(\normalsize{20}\) to \(\normalsize{1}\).A3.After an analysis much the same as the previous one, we find that \(w_1 = -\frac{5}{4}\) and \(w_2 = \frac{30}{8}\) metres per second.A4.In this case \(w_1 = -\frac{25}{11}\) and \(w_2 = \frac{30}{11}\) metres per second.

#### Maths for Humans: Linear, Quadratic & Inverse Relations

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