Mechanical mindgames

In this problem we provide a list of things to ponder concerning the mechanics of particles moving in straight lines.

Think about some which catch your attention. Some might have numerical answers; some might involve simply thinking about the mechanical issues involved. Some might require you to find extra data and others might raise more questions than they answer! They are grouped into three groups of three according to how straightforward we feel that they are to get into. Of course, you may disagree: there are no hard and fast answers to most of the questions.

 

Part I

• A fully loaded supertanker weighing $500,000$ tonnes is drifting along with a top speed of $30\textrm{ km h}^{-1}$. Estimate how many golf balls would Tiger Woods need to strike at the tanker to bring it to a stop, ignoring all friction.
• Describe some interesting real-world objects with momenta ranging from around $10^{-6}$ to around $10^6\textrm{ Ns}$. Try to find examples with momenta $M_1, M_2, \dots, M_{12}$, measured in $\textrm{Ns}$ which you know for sure lie within the power ranges as follows: $$10^{-6}< M_1< 10^{-5}< M_2< 10^{-4}< \dots < M_{11}< 10^{5}< M_{12}< 10^{6}$$
• It is common in certain films for a car to drive up a ramp onto a moving lorry. What might happen if this were tried in real life? Would it make a difference if the car were front or rear wheel drive?

Part II

• In July 1994 the Comet Shoemaker-Levy 9 struck the plant Jupiter at a speed of around $60\textrm{ km s}^{-1}$. Although it broke up before impact, the core of the original comet was around $5\textrm{ km}$ diameter. Its estimated density was $0.3 - 0.7\textrm{ g cm}^{-3}$. Imagine such a comet had struck the earth. How much would it have changed the earth's velocity relative to the sun?
• It is said that striking a pedestrian at $40$ miles per hour in your car leads to an $80\%$ chance of death whereas striking a pedestrian at $30$ miles per hour leads to a $20\%$ chance of death. Why can there not be a linear relationship between velocity and survival chance? What might a realistic graph of velocity against survival look like?
• Several cars are waiting on a dry road in stationary traffic with a distance of around $2$ metres between each car. The coefficient of friction between the cars and the road is about $0.7$. A lorry travelling at $60\textrm{ km h}^{-1}$ smashes into the rear car shunting the cars into each other. Estimate the number of cars which will be involved in the crash, being very clear about the modelling assumptions and mechanics used.

Part III

• Two spheres of solid lead of radius $1\textrm{ m}$ are in deep space stationary relative to each other and a fixed origin with a distance of $3\textrm{ m}$ between their centres. If we ignore all gravitational effects other than those due to the two spheres, estimate how fast they will be moving when they strike. You might also try to estimate how long it will take before they collide. How do the results change for $1\textrm{ cm}$ lead spheres with a distance of $3\textrm{ cm}$ between their centres?
• When a particle falls under the action of gravity, the acceleration is given by Newton's 2nd law $F=ma$ and the magnitude of the force by Newton's law of gravity $F=-\frac{GmM}{r^2}$. Experimentation indicates that the numerical value of the little $m$ in each equation is identical, and the principle of equivalence (which appers to be true) asserts that they are mathematically identical. Think about this; it is good to understand. Does it surprise you?
• A certain mathematical particle is defined to have speed $1$ for irrational values of time and speed $0$ for rational values of time. Would it make sense for this particle to move? (To consider this question, you might like to read a little about Lesbesgue integration).