Physics 5.1

Topic 5 Cars Mate !1a) Measuring Distance d and Displacement s (or r)"Distinguish between scalar and vector quantities in equations."Distance and displacement are related. Both are measured in metres.

Distance is a scalar quantity. It has no direction. Distance can twist, turn andwriggle. In a school, distance is measured with a trundle wheel that followsthe wriggly line. In a car, distance is measured using the car's odometerwhich clocks up the kilometres travelled. When writing down a scalarquantity such as distance, you record 2 things: the number and the unit.

Displacement is a vector quantity. It has direction. It is measured in a straight line from thebeginning to the end of the journey. This line should have a compass bearing which is thenumber of degrees clockwise from north. When writing down a vector quantity, you record 3things: the number, unit and direction. For example, if a man walks 50 metres due S.W., we writehis displacement as: 50 metres due bearing 225o (S.W.)

☺ ☺ ☺ ☺ ☺ Set 5.1 Measuring a Journey

Sam Thomas

X

Betsy

Herby

Bert

Each vehicle above starts from the same central point X and covers a journey indicated by acomplex path. The scale for the diagram is 1cm = 5km. Find the displacement and distancecovered for each vehicle in km during the whole journey. Watch the number of significant figuresin your answer.

Physics 5.2

Solutions 5.1Sam d = 46km, r = 31km due bearing 291o ;Thomas r = 44km, s = 38km due bearing 67o

Bert d = 35km, r = 35km due bearing 81o ; Herby r = 43km, s = 40km due bearing 119o

Betsy d = 80km, r = 21km due bearing 203o

b) We live in a world of changing speeds"Identify that a typical journey involves speed changes."

Speed is the time rate at which an object is putting distance behind it. As we are not inert beingsin the Lord's Universe, we are constantly doing things to ourselves to change our speeds. We getup, go to work, run, walk and lie down for hours at a time. What a crazy race to model usingphysics equations!

We do have devices that can measure instantaneousspeeds. Drivers have a speedometer, which uses amagnetic gismo to measure the rate of rotation of thewheels.

Police bounce radar beams off cars moving relative tothem. The movement of the target car changes thefrequency of the waves and this can be processed by acomputer to give an instantaneous speed reading.

c) Instantaneous and Average SpeedsThe instantaneous speed is the time rate of movement at aparticular point in time. We need fairly high technology tomeasure this type of speed.

The average speed is easier to compute and uses the formula

average speed = distance time

d) Velocity and Speed"Compare instantaneous and average speed with instantaneous and average velocity."Velocity and speed are also related quantities. Both are measured in metres/sec. This means thatthey both measure metres covered per second. Speed is a scalar with no direction. Velocity is avector and is directional.

For a given motion, there are 4 different quantities that can be measured that have the unitmetres/sec. These are average speed, average velocity, instantaneous speed and instantaneousvelocity. The first two are easily measured with a metre rule and stopwatch. The last two requirethe drawing of graphs and the use of calculus. A car's speedometer measures instantaneous speed(in km/hour unfortunately).

The diagram right can be described in two ways:

The car has constant speed of 15 metre sec-1

The car has constant velocity of 15 metre sec-1

due bearing 60o (measured from N)

Speedometers measureinstantaneous speed

Physics 5.3

In this diagram we have two cars travelling at the same constant speed where the wheels cover10 metres every second.

Whilst the first car is travelling at constant velocity, the second car is constantly changing itsdirection, and hence, its velocity. So velocity changes if direction changes.

e)The Average Velocity and Average Speed Formulae

Average Speed = distance = dtime t

Average Velocity = change in displacement = Δrchange in time Δt

The syllabus expects us to use the first formula in a prac and to practise the second formulabelow. So we will state the second formula:

vav = Δr Δt

Where Δr is change in displacement in metres.vav is average velocity in metre sec-1

Δt is measured time interval in secondsd is distance in metres

Note that displacement r or s isdistance measured in a straightline from the start of themotion to the end.

Distance d is longerand is measured by carodometers in km.

r

Sometimes I willuse the old fashionedsymbol s instead of rfor displacement.

Soon we will use rfor circle radius, whichI find confusing.

Physics 5.4

Basic TrainingAll simple mathematical Physics problems can be solved by following these basic steps:-

Often you can start with a sketch so that you can visualise the situation better. Then:

Step 1. Write down the symbols and values for the quantities given. Convert to M.K.S. values(eg kilometers to metres, tonnes to kilograms, hours to seconds)Step 2. Write down the symbol for what you are asked for followed by a question mark.Step 3. Write down the formula. Change the subject and substitute in.

Example The velocity of light is 300,000 kilometers per second. How far will light travel in1 minute ?

v = 300,000 km sec-1

= 300,000,000 metre sec-1

t = 1 min= 60 sec

d = ?

v = dt

d = vt= 300,000,000 x 60= 1.8 x 1010 m

"Define average velocity as:

vav = ΔrΔt"

"Solve problems and analyse information using the formula:

vav = ΔrΔt

where r = displacement"

☺☺☺☺☺ Set 5.2 Average Velocity or Average SpeedGood setting out is vital in physics. Set your calculations out in neat columns. Avoid cuttingcorners. Remember to convert all numbers to the magic metres, kilograms and seconds.Note that if a velocity is asked for, you must specify direction in your answer.

1. A car odometer registers 270km travel on a typical winding road. If the trip takes 4.00 hours,what is the average speed for the trip in M.K.S. units ?

2. How far would the same car travel at the same average speed in one minute ?

3. A particularly ugly non biodegradable wind-up toy that says “Oiy” can travel at an averagespeed of 0.200 metre sec-1 when using the more polluting alkaline cell. How long will it take totravel 200 metres ?

Physics 5.5

4. Sound travels at 340 metre sec-1. An aircraft can travel at an averagespeed of Mach 2 (twice the speed of sound) and has a range of 1500 km atthis speed. How long can it remain in the air at Mach 2 ?

5. The township of Burradulla lies exactly North West of Duckabilla . Totravel to Burradulla from Duckabilla, you have to navigate a 25.0kmwinding dirt track. It takes an hour and makes the most hardened outbacktypes say "bother" .....or something like it. The bleached crows, which flybackwards in these parts, reckon that the trip is only 12.0km in a straightline. Find for the trip :

a) the average speedb) the average velocity

6. Redex trials were famous in the 50's andwere sponsored by a brand of engine oil. Onestar rally driver was called "Gelignite Jack"because of his habit of confusing the oppositionby blowing up roadsigns. This particular roadrally, is in the shape of an an exact semi-circleof radius 500km . The finishing point is directlyeast of the starting point. Jack's mud coveredbomb averaged 20.0 metre sec-1. Determine:

a) his time taken for the rallyb) his average velocity

7. A highway speed limit is set at 60 km/hr in an attempt to prevent careless macho fools fromkilling themselves in their misplaced love affair with their chromic curves. This means that youcan do 60km in 1 hour. Calculate this speed in metre sec-1 by substituting into the speed formula.

8. How long would it take a 60 km/hr vehicle to travel 1.00 km ?

9. The town of Mopoke lies 400km north of Wallyrobber. Byroad the trip takes 9.00 hours on a winding road. Find :

a) the average velocity for the tripb) the distance by road if a car can travel at an average speed of 18.5 metre sec-1

10.Bert can pedal his penny farthing at 17.3 metre sec-1. Howlong should it take him to pedal 5.00km ?

11. Cathy Caterpillar tends to take the long and winding path.After a hard 8-hour day's foot-slooging, she finds herself backwhere she started. If her average speed was 3cm/sec find:

a) the distance travelledb) the average velocity

Physics 5.6

Solutions 5.21. 18.75 metre sec-1 2. 1125 metre 3. 1000 sec 4. 2206 sec 5a) 6.9 metres/sec b)3.3 metre sec-1 N.W. 6a) 7.86 x 104 sec b) 12.7 metre sec-1 east 7. 16.7 metre sec-1 8. 60 sec9a) 12.3 metre sec-1 North b) 5.99 x 105 sec 10. 289 sec 11a) 864m b) nil

Mandatory Prac 1 Measuring average speed"Plan, choose equipment for, and perform a first-hand investigation to measure the averagespeed of an object or vehicle."

Probably the best way to do this would be to one of these:a) Bring in a model electric train set and set it up. Work out a way of measuring d. Doseveral runs and measure t with a stopwatch. Tabulate your values. Convert all numbers tometres and seconds. Watch significant figures to ensure that you are measuring the limitsof accuracy. Calculate the individual average speeds and calculate the mean. The answershould have the least number of significant figures in the quantities used. Estimate theerror as the greatest deviation form the mean.b) Alternatively use a wind up or friction car in the playground.c) Measure a distance along a stretch of road between poles and time some cars.

f) Graphing Motions"Present information graphically of:

- displacement vs time- velocity vs time

for objects with uniform and non-uniform linear velocity."

Most motions are very complex and difficult to analyse. It is possible to calculate all sorts ofquantities from displacement/time and velocity/time graphs. Four hundred years ago, a boy calledNewton invented calculus to help us calculate these quantities. The methods of calculus involve:

a) differentiation - measuring tangent to graphsb) integration - measuring area under graphs

(i) Displacement/time graphs for uniform linear velocityThis is a displacement/time graph for the motion of Anthony Aardvark

Uniform velocity s/t graphs arestraight lines.

We can use Newton'stechnique of differentiationto calculate the velocity.

v = Δs Δt

= 5-1 8-0

= 0.5 ms-1

Physics 5.7

☺☺☺☺☺ Set 5.3 Displacement/time graphs foruniform linear velocity

These are data for the epic journeys of some of may favourite creatures.Plot the data on a displacement/time graph.Use the method of calculus to compute the velocity for the journey.Notice that I have been a very very good boy and have not left any naked numbers in the datatable. Numbers should always be followed by a unit.

1. Carl Crocodile who never smiles

time displacement0 sec 12m3 sec 18m6 sec 24m9 sec 30m

2. Bruce the Bounder from Engonia

This could be a good mate of Brucey Shillingsworthfrom Engonia. Bruce was a very good artist and agood student. Now I'm teaching all his relatives.

time displacement0 sec 0m2 sec 4m4 sec 8m6 sec 12m8 sec 16m10 sec 20m

3. Angus Ant from Ardlethon

time displacement0 sec 20m2 sec 15m4 sec 10m6 sec 5m8 sec 0m

Physics 5.8

Solutions 5.3

1.

2.

3.

Physics 5.9

(ii) Displacement/time graphs for uniform accelerationNear the surface of the earth, the falling of objects under gravity approximates uniformlyaccelerated motion. On earth when things fall, they increase their velocity by 9.8 metres/sec everysecond.

If you fall off a ladder for 1 second, you are almost travelling at the world record speed forhumans. After 2 seconds of fall, you are doing almost 80 KPH.

☺☺☺☺☺ Set 5.4 Displacement/time graphs fornon-uniform linear velocity

This is the scarey data for killer drop-bears who are known to inhabit the Horsetrailern bush.Paul Hogan fears them. So does my mad brother-in- law who tends to sleep under trees withsuch beasts who have a nasty habit of roaring all night to attract their girlfriends. They are alsoknown to do wee-wees - from a great height and that can be lethal in the Tarquine wilderness.So wear your hat Bob Brown!

Plot this data on a displacement/time graph:

Time Displacement0 sec 0 m1 sec 4.9 m2 sec 19.6 m3 sec 44.1m4 sec 78.4m5 sec 122.5m

Use the method of tangents to attempt to estimate the velocity after 2 seconds:Place an X on the graph at the 2 sec mark. Draw a tangent at this point and construct anytriangle using the tangent line as the hypotenuse. The other two sides must be vertical (s-side)and horizontal (t-side)

As before v = ΔsΔt

The correct answer is 19.6 ms-1.Were you close?

Physics 5.10

(iii) Velocity/time graphs for uniform linear velocityThis is a no-brainer. This is a velocity/time graph forTed Turtle. See if you can work out his velocity. I hopeby now you have worked out the shape of a velocity/timegraph for constant velocity motion.

There are microwave sensors which can bouncewaves off moving objects to detect velocities.They can be connected to data-loggers to tabulateand graph velocity against time.

The interesting thing about velocity/time graphs is that we can do a new process calledintegration. Integration involves calculating the area under the graph up to time t.

The area under a v/t graph is the displacement. Newton would have expressed it this way:

s = sv dt

To do this involves finding the area under the curve within time boundaries.

Eg find the displacement of Ted after 5 seconds.The answer is the area of the rectangle drawn. It cuts off at t = 5 seconds.

s = area = 3 x 5 = 15m

Dah.....whatcomes after 2 ?

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Physics 5.11

(iv) Velocity/time graphs for uniform accelerationFalling objects and masses subject to a constant force experience uniform acceleration. We nowknow the following:

a) the slope of a v/t graph gives the acceleration

a = Δv Δt

b) The area under a v/t graph gives the displacement

s = sv dt

☺☺☺☺☺ Set 5.5 v/t graphs for uniform linear acceleration1. This is a graph for a rock falling off a cliff on the planet Quetzyl a) Use the method of differentiation to find the acceleration due to gravity b) Use the method of integration to find the displacement of the falling rock after 20 seconds

2. This is a general graph for an object uniformly accelerating from initial velocity u to finalvelocity v in t seconds.a) Use the graph to find a in terms of u, v and t (The method is the same as before)b) Use the graph to find s in terms of u,v and t

Physics 5.12

Solutions 5.51a) a = Δv = 30 = 1.5 ms-2

Δt 20

b) s = area under graph which is a triangle = 1/2BH = 1/2 x 20 x 20 = 200m

2a) a = Δv = v - u Δt t

b) s = area under graph which is a rectangle plus a triangle = BH + 1/2BH = ut + 1/2 (vt - ut) = 1/2(vt-ut)

We have now derived an acceleration formula.

2. Accelerated MotionMasses which change their velocities are said to be accelerating. To change your velocity, you canchange your speed or direction or both.

"Define average acceleration as:

aav = Δv Δt

Therefore aav = v - u" t

At last we have a simple formula without these wretched deltas. All measured times, distancesand displacements are deltas anyway as they are relative. It probably even applies to masses andenergy.

Notice that acceleration is ..........change in velocitytime

Where a is acceleration in metre sec-2

u is initial velocity in metre sec-1

v is initial velocity in metre sec-1

t is time in seconds

Physics 5.13

When cars change their velocity, they are accelerating. Cars accelerate when the accelerator isapplied, when the brakes are applied, when there is wind or tyre resistance, a collision or whenthe direction of the car changes. Accelerations are produced by forces on the car. When youaccelerate, the passengers feel it. You might like to think of all the different forces that can act onthis removalist van below to produce the accelerations. Forces are vectors and are represented byarrows. They point in the direction of the acceleration or change in velocity.

Motion, where the acceleration constantly changes, is hard to model. Uniformly acceleratedmotion is easier to model . The exercise will develop your ability to apply equations properly.

a) Uniformly Accelerated Motion

This is motion where the velocity changesby the same amount each second. Uniformacceleration is produced when a constantforce is applied to a mass.

This is the case when an object falls nearthe surface of the earth under gravity. Theacceleration produced by earth gravityforce is 9.8 metre sec-2

and is sometimesgiven the symbol g.

For all motions

Acceleration = Change in velocityChange in time

a = Δv where Δ means change inΔt

For constant acceleration a = v - u t

Where v is final velocity and u is initial velocity

Equations of Uniformly Accelerated MotionThe equations of motion are famous in physics. They work only for constant acceleration only .You will use derivatives of these equations for the year 12 course projectile motion section.

v = u + at ...... (i).... no s

s = ut + 1/2at2 .... (ii).... no v

v2 = u2 + 2as .... (iii)... no t

The first equation is part of the year 11 syllabus.The final worksheets will practise using the other formulae as well and are extensions for uni.Your choice.

s is displacement in metrest is time in secsu is initial velocity in metre sec-1

v is final velocity in metre sec-1

a is acceleration in metre sec-2

Physics 5.14

Example A car travelling on the expressway begins with a velocity of 20 metre sec-1 and strikesa headwind when an asteroid hits the Pacific. After 5 seconds it is travelling in the reverse direc-tion at 10 metre sec-1 . Find the acceleration.

Step 1 Always select a positive direction first and stick to it !Step 2 Write the symbols s u v a t in a columnStep 3 Write the 3 things that you are given and ? next to what you wantStep 4 Choose a formula and substitute

Choose + ve to be the initial directionsu = 20 metre sec-1

v = -10 metre sec-1

a = ?t = 5 sec

v = u + ata = v - u

t= -10 - 20

5= -6 metre sec-2

Never forget the units and show full setting out. You need 3 given quantities to work out theanswer.

☺☺☺☺☺ Set 5.6 First Equation of Motion v = u + atIn each problem 3 things are given. Remember this. Some of the three things may be hidden inthe language of the question.

1. A spacecraft moving at 500 metre sec-1 does a 20 metre sec-2 burn for 20 seconds. Find thenew velocity if the acceleration is forward.

2. A spacecraft is moving at 500 metre sec-1 and uses reverse thrust braking rockets for 1 minuteat 20 metre sec-2. Find the new velocity.

3. The brakes of a car travelling at 20 metre sec-1 cause a negative acceleration of 5 metre sec-2.How long will it take the car to stop?

You vill need Suvat !List Suvat symbols ina column. Be neat .

Physics 5.15

4. A tough fighter pilot can withstand 100 metre sec-2 acceleration without blacking out. What isthe minimum time needed for her to accelerate her aircraft to Mach 2 (Mach 1 = 340 metre sec-1)from a cruising speed of 200 metre sec-1 ?

5. The acceleration due to gravity near the surface of the earth is 9.8 metre sec-2. A parachutistfreefalls for 12 seconds before opening his “chute”. What is his velocity when this happens ?

6. How long can a parachutist freefall from rest before hitting the speed of sound ?

7. A student rebounds on a trampoline at 4.9 metre sec-1 . How long does it take to reachmaximum height ?

8. A car moving at 15 metre sec-1 brakes and loses half its speed before impact with a mad schoolstudent running the lights on the school crossing. If the time of braking is 2 seconds, what is thebraking deceleration ?

☺☺☺☺☺ Set 5.7 Equations of Motion - An extension assignment only1. A car travelling at 20 metre sec-1 brakes uniformly to a halt in 2 seconds. Calculate theacceleration and use the acceleration to find the length of the skidmarks.

2a) A car travelling at 15 metre sec-1 hits a pole so that the front of the carcrumples 0.5 metre. Find the acceleration experienced by the driver. b) Repeat if the car is travelling twice as fast.

3. A car falls off a 98 metre cliff, starting from rest.a) How long will it fall ?b) What will be its velocity at the base ?

4. A dysfunctional kangaroo leaves the ground vertically with an initial velocity of 9.8 metre sec-1

.a) How high will it leap ?b) What will be its velocity when it returns to the ground ?c) How long will it take to return to the ground ?

5. A vehicle moving at 10 metre sec-1 accelerates at 2 metre sec-2 for 4 seconds. What distancewill it cover during this time ?

6. What acceleration is required to increase the speed of a car from 10 metre sec-1 to 20 metresec-1 in a 100 metre stretch of road ?

7. A child slides from rest down a slippery dip 10 metres long. Its acceleration down the 30degree incline is half the acceleration due to gravity. How long will the slide take ?

8. A car travelling at 25 metre sec-1 stops by collision in 0.1 seconds. What is the deceleration ?

9. A car travelling at 20 metre sec-1 decelerates to5 metre sec-1 in 15 seconds. What is :-a) the acceleration ?b) the distance covered covered during the time interval ?

Physics 5.16

10. A space vehicle moving at 600 km/sec fires its rockets for 1 hour and finishes with a velocityof 300 km/sec in the same direction. Find :-a) the accelerationb) the displacement of the space vehicle during this time

Solutions 5.61. 900 metre sec-1 2. -700 metre sec-1 3. 4 sec 4. 4.8 sec 5. 117.6 metre sec-1 6. 34.7 sec 7.0.5 sec 8. -3.75 metre sec-1

☺☺☺☺☺ Set 5.8 Extension assignment onlyMore Problems Involving Acceleration in 1-D

1. A tugboat moving at 20 metre sec-1 hits a rock andgrinds to a halt over a distance of 10 metres. Find the

a) acceleration of the tugboatb) stopping time

2. A “ Freedom Fighter” feels like “Freedom Fighters”often do and celebrates the revolution by firing his gunvertically. The muzzle velocity is 490 metre sec-1.

a) How long is it before the intrepid warriordiscovers that “what comes up must come down” ?

b) What will be the velocity of theprojectile when it comes back down?

3. A good high jumper can jump 2 metres.a) What is the vertical jump off velocity ?b) How long for the jump? (up and down)

4. A ship moving at 20 metre sec-1 loses a man overboard. It applies a braking acceleration of -0.1 metre sec -2. Calculate

a) the time taken for the ship to stopb) the time taken for the ship to return to the swimmer beginning at the timeof the accident.c) the velocity of the ship when it returns.

5. A lookout hangs 490 metres above a valley floor. An empty tourist bus suffers brake failure androlls through the safety fence. It falls vertically.

a) How long will it fall?b) What will be its velocity at the base.

6. A space shuttle chases MIR (the russian space lab) at 2000 metre sec-1 relative to MIR. It is8000 km away. It needs to be at rest at the time of docking and begins a constant decelerationburn.

a) What deceleration is necessary using the brake jets ?b) How long until rendezvous ?

Physics 5.17

Solutions 5.71. a = -10 metre sec-2, 20 metres 2a) -225 metre sec-2 b) -900 metre sec-2 (4 times the otheracceleration when travelling twice as fast) 3a) 4.47 sec b) 43.8 metre sec-1 4a) 4.9 metre b) -9.8metre sec-1 c) 2 sec 5. 56 metre 6. 1.5 metre sec-2 7. 2.02 sec 8. -250 metre sec-2 9a) -1ms-2 b)187.5 metre 10a) -83.3 metre sec-2 b) 16.21 x 108 metre

Solutions 5.81a) -20 metre sec-2 (roughly 2g) b) 1 sec 2a) 100 sec b) -490 metre sec-2 3a) 6.26 metre sec-1 b)1.28 sec 4a) 200 sec b) 400 sec c) -20 metre sec-1 5a) 10 sec b) 98 metre sec-1

6) -0.25 metre sec-2 b) 8000 sec