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**K M Cheng Department of Physics, CUHK Time Allocation: 6 hours**

Science and the Road K M Cheng Department of Physics, CUHK Time Allocation: 6 hours

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**Content Measuring motion Speeding up and slowing down**

Tyre/road friction（摩擦力） Stopping a car The speed formula Collision Travelling round a bend Driving safety

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**Measuring motion Speed（速率）tells us how fast an object moves.**

The speed at a particular instant of time is found by taking the average speed over a very short time interval. Velocity（速度）tells us the speed of the object as well as its direction of travel.

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An object accelerates when its speed increases and decelerates when its speed decrease. Acceleration（加速度）is the rate of change of velocity. For motion in a straight line, When t becomes very small, the average acceleration approaches the acceleration at a particular time instant.

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**For uniformly accelerating motion in a straight line:**

where

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**From these two expressions, the three equations of motion can be deduced:**

Note that s is the displacement（位移） which is the distance travelled along the straight line.

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**Speeding up and slowing down**

We may change the motion of a moving object by pushing or pulling it. Scientists call this pull or push to be a force（力）. The unit of force is newton (N - 牛頓). Newton formulated three laws of motion describing the properties of force. In Newton’s first law, it states that: A body continues in a state of rest or uniform motion in a straight line, unless an unbalanced force acts on it.

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**What does the term unbalanced force really mean?**

These forces balance. The block does not move. These forces does not balance. The block moves as if it was being pulled by a force of 10 N to the right.

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Newton's first law of motion tells us that all objects resist change in their state of motion. All objects have this tendency, i.e. they have inertia（慣性）.

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**The mass（質量）of an object is a measure of its inertia**

The mass（質量）of an object is a measure of its inertia. An object with a larger mass has more inertia and thus has larger tendency to resist changes in its state of motion. The mass of an object is usually given in unit of g (gram) or kg (kilogram).

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**If an object is dropped, it falls to the ground at increasing speed**

If an object is dropped, it falls to the ground at increasing speed. The free falling object（自由落體）accelerates due to the pull of the Earth’s gravity（重力）. When air resistance is negligible, all free falling objects fall with the same uniform acceleration (approximately) known as the acceleration due to gravity g. The value of g is independent of the mass of the body, and is usually taken as 10 m/s2.

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**A car will eventually stop if the engine is turned off**

A car will eventually stop if the engine is turned off. It is due to the friction（摩擦力）which slows down the cars’ motion. Friction arises whenever an object slides or tends to slide over another object. It always acts against the motion. PUSH To Move FRICTIONAL FORCE to Resist Motion

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**Friction is due to the irregularities on a surface**

Friction is due to the irregularities on a surface. Even a smooth surface has these “peaks and valleys”. If two surfaces rub over each other, the irregularities on the two surfaces would crush together, resulting in an opposing force.

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**What does friction depend on?**

Friction does not depend on the area of contact, but depends only on the nature of the contact surface. Friction is much reduced if the contact surface is lubricated with water or oil, and if rollers and bearings are placed between the contact surface. The greater is the weight of the object, the greater is the friction.

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Tyre/road friction According to Newton’s Third Law, the car is driven by the forward frictional push of the ground on tyres. If we apply the brakes (very hard), the wheels are locked and thus prevent from rotating. As a result, the car skids and decelerates. The decelerating force is actually the sliding friction（滑動摩擦）– a backward push of the ground on tyres. Anti-lock braking system (ABS - 防鎖死煞車系統 ): a system on motor vehicles which prevents the wheels from locking while braking.

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**Tyre/road friction On a level road, this force is given by**

where is the coefficient of tyre/road friction and mg is the weight of the car. The value changes only slightly with speed. However, if the road surface is wet, the value depends significantly on tyre conditions, speed, weight of vehicle and degree of wetness.

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**Tyres and tread patterns**

Various tread patterns are designed to provide a firm grip on the road surface Groove on the thread are for pushing out water on a wet road so that the tyres and the road are still in direct contact and are not lubricated Tread pattern on a car tyre Tread pattern on a mountain bicycle tyre (Wikimedia commons)

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**Tyres and tread patterns**

Worn-out tyres are dangerous and illegal By law, tread depth must not be less than 1 mm over three-quarters of the tread area A penny can be used to check tyre tread (Penny test, USA): Take a penny and put Abe's head into one of the grooves of the tyre tread If part of his head is covered by the tread, you're driving with the legal amount of tread If you can see all of Abe's head, it's time to replace the tyre (Wikimedia commons)

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**Slick tyre A type of tire that has no tread pattern**

Used mostly in auto racing By eliminating any grooves cut into the tread, such tyres provide the largest possible contact patch to the road Can be very dangerous if the racing track is wet! (Wikimedia commons)

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The skid marks When a car skids, particularly on a tarmac road surface, clearly visible skid marks are usually left on the road The speed of a vehicle prior to skidding can be estimated from the skid mark length and the coefficient of friction（摩擦係數 ） between the tyres and the road surface (Wikimedia commons)

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**Skidding over different surfaces**

Very often, a skid mark may extend over different surfaces, e.g., tarmac（柏油碎石）, concrete（混凝土）, gravel（砂礫）, grass, etc. In these cases, the speed of a vehicle prior to skidding can be estimated from the skid mark lengths on different surfaces and the coefficient of frictions between the tyres and different surfaces

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Skidding on slopes When the road surface is not level, the skidding distance is affected. It will be shorter for going uphill and longer for going downhill. One may define the gradient e（斜率）of a non-level road by H L

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Skidding on slopes If the coefficient of frictions of a level surface is , then the effective value for an up-slope or down-slope of gradient e and the same road type is approximately given by Up-slope: Down-slope:

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**Skidding on slopes Proof of the formulae: if is small N = mgcos N**

mgsin N N = mgcos H L if is small Force diagram for the case of down-slope

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Stopping a car When a driver sees a hazard, he hits the brake to stop the car. The time taken for the driver to react is called the reaction time (or thinking time). Reaction time varies from person to person but is, on average, about 0.2 s. The distance travelled by a vehicle during reaction time is called reaction distance, i.e.

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**Measuring reaction time by the ruler drop technique **

Your partner holds a ruler between your fingers. The ruler is released suddenly. As soon as you see it, close your finger to catch the ruler. Record how far the ruler drops before you catch it. The faster you react, the less the ruler will drop before being caught. Remark: the reaction times are much longer in the case of unexpected events – in the range 1.5 to 2.5 seconds!

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**Once the driver applied the brakes, the car slows down**

Once the driver applied the brakes, the car slows down. The distance that the car moves during braking is called the braking distance (can be estimated by the skid mark length). Stopping distance is the distance taken to stop a car, i.e. And the time taken to stop a car is called the stopping time.

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**Example Thinking time = 0.75 s**

0 m/s 12 m/s 0.75 s 2.5 s Stop signal Starts to brake Slowing down Stopped time Speed Thinking time = 0.75 s Distance Travelled during thinking time = 12 m/s 0.75 s = 9 m Distance Travelled during braking = ½ 2.5 s 12 m/s = 15 m Total stopping distance = 9 m + 15 m = 24 m

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**The stopping distance depends on the following factors:**

the speed of the car the driver’s reaction time the road surface the condition of the car’s brakes and tyres

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The speed formula Suppose a vehicle of mass m travel on a level road at a speed u prior to skidding. The kinetic energy of the vehicle is given by s u u = 0

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The speed formula After skidding a distance s, the vehicle stops. The decelerating force is the tyre/road friction and is given by The work done W against friction is s u u = 0

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The speed formula When the vehicle stops, its kinetic energy is reduced to zero. The change of kinetic energy is equal to the work done against the friction, i.e. OR So the skid-to-stop distance depends only on the coefficient of tyre/road friction () and the speed of the vehicle (u) prior to skidding.

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Some Learning Points In braking , a car’s kinetic energy (KE) must be dissipated (usually as heat) KE = work done in stopping the car = mgs Braking distance is proportional to the speed squared The effective coefficient of friction is not the same for all cars The coefficient of friction decrease by about 40% in wet conditions (assuming the tyres are in good condition)

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Collision Traffic accidents almost always involve collisions of one kind or another The principle of conservation of linear momentum（動量守恆）can be applied: The total momentum of the objects before collision is equal to the total momentum after collision, provided that there is no external force acting on the colliding objects. (Newton’s First Law)

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Collision The momentum of an object is defined as the product of its mass m and velocity v. For two colliding vehicles of masses mA and mB travelling along the same straight path, their velocities are uA and uB before impact and vA and vB after impact. Then the total momentum before and after impact is related

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Collision (remarks) Momentum is a vector quantity. If the collision occurs at an angle, the vector sum of the momentum before impact is equal to that after impact. Vehicles exert force on one another on impact (Newton’s Third Law). The force of impact causes damage to the vehicles and changes their velocities (and hence momenta). In fact, the force of impact, F, is related to the momentum, P, by Newton’s Second Law!

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**Travelling round a bend**

A vehicle travelling round a bend on a level road can be viewed as moving along a circular path The centripetal force（向心力）is pointing towards the centre of circular motion Centripetal force Centre of circular path r v

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**Travelling round a bend**

The centripetal force is provided by the sideway friction between the tyres and the road surface: Equating the two forces, we get v is called the critical curve speed for the bend

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**Travelling round a bend**

When the speed of the vehicle is smaller than the critical curve speed of a bend, the vehicle has no difficulty in negotiating the bend. When the vehicle is just at the critical curve speed, it is travelling at the limit of adhesion to the road. It cannot brake or steer onto a tighter course to avoid an unexpected hazard without risking side-slipping. When the speed of the vehicle is greater than the critical curve speed, the frictional force is not large enough to provided the necessary centripetal force. As a result, the vehicle side-slips.

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**Travelling round a bend**

If the bend is banked, the vehicle may negotiate a bend at a higher speed because a component of the normal reaction contributes to the centripetal force as shown:

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**Travelling round a bend**

When a bicycle (or motorcycle) travel round a bend, it is necessary to lean it into the turn. Otherwise the centrifugal forces will throw the bicycle (or motorcycle) over on its side.

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**Driving safety Major reasons of car accidents [3]:**

Following too closely to vehicle in front. Losing control of vehicle. Careless lane changing. Improper or illegal turn. Starting negligently.

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Many people are injured or even killed by traffic accidents in each year. Whether as pedestrian, driver or passenger, we should be aware of road safety to avoid traffic accidents. Almost all modern cars are equipped with some safety features which can help to protect the driver and passengers during an impact (see next page figure).

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**Don’t drink and drive! Alcohol Affects Your Driving Ability**

Impairs judgment of speed and distance Slows down reaction time Affects the co-ordination of the body's movements Blurs vision Gives a false sense of confidence

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**Don’t drink and drive! The effect of blood alcohol content (BAC)**

No. of standard drinks BAC (mg/100 ml) Effects 1 noticeable effects (on perception etc.) 5 100 to 150 intoxication 24 240 to 360 unconsciousness 36 360 to 480 coma or death BAC < 50 ml/100 ml under the law in Hong Kong!

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References: G. Bethell and D. Coppock, Physics first. Oxford: Oxford University Press, 1999. G. Alderton, D. Berrington, and M. Brimicombe, Revise for GCSE Science: Salters. Oxford: Heinemann, 1999. P. K. Tao, The Physics of Traffic Accident Investigation: Oxford University Press, 1987

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Example 1 A car skids with all four wheels locked and leaves skid marks of 19.3 m on a tarmac road surface, 5.6 m on a concrete pavement and 15.4 m on grass. The coefficients of friction of the tarmac, concrete pavement and grass are 0.74, 0.82 and 0.46 respectively. Estimate the speed of the car at the start of skidding.

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Example 1 The kinetic energy of the skidding car is equal to the total work done against friction over different surfaces. Hence where 1, 2, 3 are the coefficients of friction of the tarmac, concrete pavement and grass respectively.

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Example 1 The speed of the car at the start of skidding is given by

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Example 2 A car skids with all four wheels locked for 38.5 m and then runs into a tree. The impact speed of the car is estimated from the damage to be 40 kmh-1. The coefficient of tyre/road friction is found to be Estimate the speed of the car prior to skidding.

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Example 2 The car decelerated on skidding. The decelerating force F is given by If the average deceleration over a skidding distance is a, then (Newton’s Second Law)

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Example 2 Applying the equation of uniform acceleration,

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Example 3 A car skidded 12.6 m before hitting a parked van on its side. The two vehicles became locked together and skidded a distance of 3.6 m before coming to a stop. The coefficient of tyre/road friction was found to be Total mass of the car and its passengers was 1,260 kg and total mass of the van and its load was 2,100 kg. What was the speed of the car before impact? What was its speed at the start of skidding?

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Example 3 Since the car and the van were locked together after impact, they had a common post-impact velocity v. From the skid mark, we get

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**Example 3 The pre-impact speed of the car**

Applying the principle of conservation of momentum, we have The pre-impact speed of the car

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Example 3 The speed of the car at the start of skidding is

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