Download presentation
Presentation is loading. Please wait.
1
GCSE Compound Measures & Rates of Flow
Dr J Frost @DrFrostMaths Last modified: 21st March 2018
2
Starter :: Speed-Distance-Time
A car travels at 7 m/s. How much time does it take to travel 35m? If 7m travelled each second, then 35m travelled in 5 seconds. ππ π =π seconds ? Jessica Ennis ran her 0.8km Olympic heptathlon race in 2012 in a time of 2min 8.65s. What was her average speed in metres per second? (Hint: convert quantities to metres and seconds first) 0.8 km = 800 m 2min 8.65s = πΓππ +π.ππ=πππ.ππ s πππππ
= π
πππππππ ππππ = πππ πππ.ππ =π.ππ m/s ? π π π‘ π = π π‘ π π π‘ π‘= π π In Physics you may have used a βsdtβ triangle (letters from left to right) which helps us relate speed π , distance π and time π‘: π π π‘ π π π‘ π=π Γπ‘
3
Quickfire Questions π π π‘ ? ? ? ? ? ππ/β Speed Distance Time 2m/s 10m
0.2km/s 50km 250s ? 5km/h 10km 7200s ? 36 km/h 60km 1 hr 40 min Fro Tip: Notice how the unit gives us the formula for speed: ππ/β The same applies with formulae for density and pressure. Method 1: 1 hr 40 min = 100 mins π = =0.6 km/min = 36 km/hr Method 2: 1 hr 40 min = hr π = = 36 km/hr distance over time You go 60 times further in an hour than in a minute. This time we kept the time in hours.
4
Further Examples Jeremy Clarkson travels 250 miles at a constant speed of 100mph. How much time has elapsed, in hours and minutes? π= πππ πππ =π.π hours = 2 hr 30 minutes ? To convert from hours to hours and minutes, multiply the decimal part by 60. 0.5Γ60=30 mins. On a calculator, you can use the Β° β² β²β² key. BEWARE: 3.45 hours wouldnβt mean 3hrs 45mins Getting your units right! Hans Solo and Chewbacca activate their warp drive and travel at the speed of light. Light travels at a speed of km/s. What distance have they travelled after a year? Choose ο» 6,460,000,000,000,000m οΌ Choose 9,460,000,000,000,000m ο» Choose 12,460,000,000,000,000m ο» Choose 15,460,000,000,000,000m
5
Test Your Understanding
1 2 ? 1 hr 20 mins = 80 mins π= ππ ππ =π.πππ km/min =ππ.π km/h ? 45.5 ? 46.0
6
Density Density gives us a measure of how tightly packed matter is within a space. The density of a rock is 2.3 g/cm3. Work out the mass of a piece of this rock with a volume of 20Β cm3. This is a rock. ? Intuitively, if there is 2.3 g for each cm3, then for 20cm3, mass must be π.πΓππ=ππ π Test Your Understanding: π ππππ ππ‘π¦= πππ π π£πππ’ππ π π£ ? 19.3 Again, the unit of g/cm3 allows you to work out the formula for density if you forget.
7
Pressure ? ππππ π π’ππ= πΉππππ π΄πππ
Pressure is how concentrated a force is on a surface. Suppose I press two different objects against my hand, but with the same forceβ¦ The first is going to hurt a lot more because the force is concentrated over a much smaller area. The pressure is therefore greater. ππππ π π’ππ= πΉππππ π΄πππ Units: Pascals is Newtons per square metre (N/m2). Pounds per square inch, shortened to psi, is typically used as the unit of pressure underwater. In the Mariana Trench, the deepest point on Earth, the pressure is psi; thatβs like a 1000 stone truck placed on each square inch! =4 ?
8
Exercise 1a Questions on provided sheet. A silver ring has a volume of 3 cm3 and a mass of 36 grams. Find the density of the silver. 12g/cm3 A force ofΒ 108Β Newtons is applied across an area ofΒ 4.5Β m2. Work outΒ the pressure. 24N/m2 Gold plating costs Β£6 per cm2. How much will it cost to plate a rectangular lid of dimensions 10cm by 24cm? Β£1440 An arctic hare runs a distance of 216km in 9 hours. How fast does it run? 24 km/h A penguin flies 4 miles in 15 minutes. What is its speed in mph? 16 mph Find the distance travelled: (a) 65 mph for 2 hours miles (b) 5 m/s for 1 minute m In the 2012 Olympics Usain Bolt won the 100m in 9.63 seconds and the 200m in seconds. For what event was he faster? Speeds: 10.38m/s vs 10.35m/s. Faster for 100m. Find the density of a metal if 100π π 3 weighs 800 grams. 8 g/cm3 [Edexcel IGCSE June2010-4H Q4a] Rosetta drives 85 kilometres in 1 hour 15 minutes. Work out her average speed in km/h. 68 km/h [Edexcel IGCSE May2013-4H Q3] Yoko flew on a plane from Tokyo to Sydney. The plane flew a distance of 7800 km. The flight time was 9 hours 45 minutes. Work out the average speed of the plane in kilometres per hour. 800 km/h 8 1 ? ? 2 9 ? ? 3 ? 10 ? ? 4 More questions on next slideβ¦ ? 5 ? 6 ? 7 ?
9
Exercise 1a ? ? ? ? 11 N1 N2 12 [Edexcel GCSE June2014-1H Q14]
Emily is driving in France. She sees this sign. Emily is going to drive to Dijon. She plans to drive at an average speed of 50 miles per hour.Β Work out how long it should take Emily to drive to Dijon. [IMC 2003 Q16] After a yearβs training, Minnie Midriffe increased her average speed in the London Marathon by 25%. By what percentage did her time decrease? 20% [JMC 2011 Q21] Gill leave Lille by train at 09:00. The train travels the first 27km at 96 km/h. It then stops at Lens for 3 minutes before travelling the final 29km to Lillers at 96 km/h. At what time does Gill arrive at Lillers? 9:58 11 N1 ? N2 ? ππ πππ=ππ ππ/π π= πππ ππ =π hours [Kangaroo Grey 2015 Q6] A cyclist rides at 5 metres per second. The wheels of his bicycle have a circumference of 125 cm. How many complete turns does each wheel make in 5 seconds? 20 turns ? 12 ?
10
Density involving Volumes
Sometimes we have to work out the volume of the solid ourselves, before subsequently working out density or mass. Recall that the volume of a prism = area of cross-section Γ length ? ππππ’ππ= 12Γ5 2 Γ15=πππ cm3 πππ π =450Γ6.6=ππππ π ? Fro Tip: Ensure any units are consistent with the others. If the unit of density is g/cm3, then your mass will be in grams.
11
Test Your Understanding
? Cross-section: 2Γ7 + 5Γ2 =ππ cm2 Volume =24Γ200=ππππ g/cm3 Mass =4800Γ8=πππππ g ? ? Warning! The units used in the diagram (cm) and the unit of the length of the prism (m) are different. The density is given in g/cm3, so what unit should we use for all lengths?
12
Exercise 1b ? ? 1 2 3 metal bars Wood: 1.11 g/cm3, Water: 1.03 g/cm3
[Edexcel GCSE June2016-2H Q14 Edited] The diagram shows a metal bar in the shape of a prism. The length of the metal bar is 120 cm. The cross section of the metal bar is shown below. All corners are right angles. The metal bar is made from steel with density 8 g/cm3. Sean has a trolley. The trolley can carry a maximum mass of 250 kg. How many metal bars can the trolley carry at the same time?Β [Edexcel GCSE(9-1) Mock Set 1 Autumn H Q12 Edited] The diagram shows a piece of wood in the shape of a cuboid. The piece of wood is 3 cm by 20 cm by 1.2 m. The mass of the piece of wood is 8 kg. The piece of wood will float in sea water if the density of the wood is less than the density of the sea water. In a large pool, 1 litre of sea water has a mass of 1030 g. Will the piece of wood float in this pool? Justify your answer. [Note: 1 litre = 1000 cm3] 1 2 ? 3 metal bars ? Wood: 1.11 g/cm3, Water: 1.03 g/cm3 1.11 > 1.03 so wood will sink (i.e. not float)
13
Exercise 1b ? ? 3 4 6kg 302 g [Edexcel GCSE Nov2011-3H Q16]
The diagram shows a solid prism made from metal. The cross-section of the prism is a trapezium. The parallel sides of the trapezium are 8 cm and 12 cm. The height of the trapezium is 6 cm. The length of the prism is 20 cm. The density of the metal is 5 g/cm3. Calculate the mass of the prism. Give your answer in kilograms. [Edexcel GCSE June2008-4H Q13b] A solid cylinder has a radius of 4 cm and a height of 10 cm. The cylinder is made from wood. The density of the wood is 0.6 grams perΒ cm3. Work out the mass of the cylinder. Give your answer correct to 3 significant figures. 3 4 ? 6kg ? 302 g
14
Combining parts of a journey
Sometimes a journey may consist of multiple parts. Given the large amount of information to process, you may find it helpful to arrange it in a table. Use the rows for speed, distance and time, and the columns for each leg of the journey: F β G G β H Speed 40 mph 54 mph Distance 10 miles 18 miles Time 1/4 hr 1/3 hrs ? ? ?
15
Considering total time/distance
Pippin the cat runs to the end of my garden at 10mph and back at 20mph. What was her average speed? Note: We werenβt given the distance! Letβs set the distance arbitrarily for now so that the numbers are easy (e.g. 20 miles). It wonβt affect the final answer. 10mph 20mph Click for Catimation FROST MANOR Period 1 Period 2 Overall Speed 10 mph 20 mph ? Distance 20 miles 40 miles Time 2 hours 1 hour 3 hours Letβs add a 3rd column that represents the whole journey. Note that the βoverall distanceβ is the sum of the distances, and the βoverall timeβ is the sum of the times. The speeds however canβt be added in this way. ? ? ? ? ? ? ? Speed = 40 3 = mph ?
16
Considering total time/distance
(For more adventurous classes/students!) We could also do it algebraically to keep the distance general... 10mph 20mph Period 1 Period 2 Overall Speed 10 mph 20 mph ? Distance π miles 2π miles Time π 10 hours π 20 hours 3π 20 hours Speed = ππ
ππ
ππ = πππ
ππ
= ππ π mph ? ? ? ? ? ? ? ?
17
Combined Density π π π£ ? ? ? ? ? ? Density of B = 1.03 g/cm3 (to 2dp)
The same βtableβ approach can also be used for a mixture of substances... Liquid A Liquid B Combined (C) Density 1.42 1.05 Mass 9.94 128.66 138.6 Volume 7 125 132 ? ? ? ? ? Density of B = 1.03 g/cm3 (to 2dp) ? π π π£
18
Test Your Understanding
Papa Smurf walked 10m at 5 m/s from A to B, and then 5m at 10m/s from B to C. What was Papa Smurfβs average speed across the journey? 1 A to B B to C Overall Speed 5m/s 10 m/s 6 m/s Distance 10m 5m 15m Time 2 seconds 0.5 seconds 2.5s ? ? ? ? ? Bob the Builder needs to travel to a building site. He drives the first part of the journey of 10 km at 20 km/h. The total journey time is 1 hour. What speed did he drive for the remainder of the journey if his average speed across the whole journey was 40 km/h? 2 1st 2nd Overall Speed 20 km/h 60 km/h 40 km/h Distance 10 km 30 km 40 km Time 0.5 hr 1 hr ? ? ?
19
Exercise 2 [IMC 2009 Q19] Driving to Birmingham airport, Mary cruised at 55 miles per hour for the first two hours and then flew along at 70 miles per hour for the remainder of the journey. Her average speed for the entire journey was 60 miles per hour. How long did Maryβs journey to Birmingham Airport take? A 6 hours B hours C 4 hours D hours E 3 hours [Edexcel GCSE(9-1) Mock SetΒ 2Β Spring 2017Β 1H Q9] 1 litre of a liquidΒ π·Β has a mass ofΒ πΒ grams. 1 litre of a liquidΒ πΈΒ has a mass ofΒ πΒ grams. A liquidΒ πΉΒ is made by mixing a volume of liquidΒ π·Β with a volume of liquidΒ πΈΒ in the ratioΒ 3:7 Find an expression, in terms ofΒ πΒ andΒ π, for the mass of 50 litres of liquidΒ πΉ. πππ+πππ [IMC 2002 Q17] I walk to the bike shop at 3 miles per hour and cycle back along the same route at 12 miles per hour. What is my average speed, in miles per hour, for the time I am actually travelling on the route? 4.8 mph [Edexcel GCSE Jun2015-2H Q16] Liquid A has a density of 0.7 g/cm3. Liquid B has a density of 1.6 g/cm3. 140 g of liquid A and 128 g of liquid B are mixed to make liquid C. Work out the density of liquid C. 0.957 g/cm3 Catniss runs the first lap of a race at a speed of 10m/s. The second lap she changes speed, and takes 40 seconds in this lap. Determine the speed she needs to go in the second lap in order to achieve an overall speed of 20 m/s with an overall time of 60 seconds. 25 m/s [Edexcel GCSE(9-1) Mock Set 2 Spring H Q9] The densities of three metal alloys,Β π΄,Β π΅Β andΒ πΆ, are in the ratio Β Β Β Β Β 13:15:21 1 m3Β of alloyΒ π΅Β has a mass of 8600 kg. Work out the difference between the mass of 5 m3Β of alloy A and 3 m3Β of alloy C. Give your answer correct to 3 significant figures. 1146kg 4 1 ? 2 5 ? 3 ? 6 ? ?
20
Exercise 2 [Cayley 2015 Q1] A train travelling at constant speed takes five seconds to pass completely through a tunnel which is 85m long, and eight seconds to pass completely through a second tunnel which is 160m long. What is the speed of the train? Note that the train enters the tunnel when the front enters, but exits the tunnel when the back of the train leave. Let the length if the train be π metres. Speed in first tunnel: ππ+π π Speed in second tunnel: πππ+π π But the train is going at a constant speed: ππ+π π = πππ+π π π ππ+π =π πππ+π π=ππ So speed of train = ππ+ππ π =ππ m/s N1 [JMO 2009 B3] Tom left a motorway service station and travelled towards Glasgow at a steady speed of 60mph. Tim left the same service station 10 minutes after Tom and travelled in the same direction at a steady speed, overtaking Tom after a further 1 hour 40 minutes. At what speed did Tim travel? When Tim catches Tom, Tom has travelled for 1 hour 50 minutes. Distance = ππΓπ π π =πππ miles Thus for Tim: π=π π π , π
=πππ, π= πππ π π π =ππ mph N2 ? ?
21
Super Challenging Extension Question
We can use this table approach for exceedingly difficult Olympiad problemsβ¦ Monday Tuesday Wednesday Walk Run Speed π 2π Distance 2π π‘ π π‘β2π 4π π‘β8π Time 2π‘ π‘ π‘β2 2π‘β4 ?1 ?2 ?4 ?5 ?7 ?8 ?3 ?6 Using the fact the total distance must be the same on Monday and Tuesday, we can work out π‘: πππ+πππ=ππβππ+πππβππ π=ππ And therefore the total distance: πππ=ππΓππ=πππ And therefore the time on Wednesday: πππ π =ππ minutes If time on Monday was 3π‘, time on Tuesday was 3π‘β6 (using information given). Splitting this in the ratio 1:2, we get the times as indicated. ?9 ?10 ?11
22
Rates of Flow Questions
ROUND 1: Mr T fills a cuboid container with the dimensions shown. If he fills it at a constant rate of 25 cm3 per second, how long will it take him to completely fill the container? 50cm Volume: πΓπΓππ=ππππ cm3 Time: ππππ ππ =ππ seconds 5cm ? 8cm ?
23
Rates of Flow Questions
ROUND 2: Mr T builds a T shaped container using two cuboids, with dimensions as pictured. Mr T pours in water at a constant rate. In 9 hours the water level reaches 45cm from the bottom of the container, as shown. Determine the time to completely fill the tank. The key to solving these questions is: Work out the rate (cm3/hr or cm3/s) at which the container fills, using the time given and the volume of liquid poured in in this time. Use this rate in relation to the total volume. This might be working out the total volume given the total time, or vice versa. 50cm 15cm 40cm 40cm Volume filled in 9 hours: ππΓππΓππ + ππΓππΓππ =πππππ cm3 Rate at which container fills: πππππ π =ππππ cm3/hr Total volume: ππΓππΓππ + ππΓππΓππ =πππππ cm3 Therefore time to fill: πππππ ππππ =ππ π π hrs ? 30cm ? ? 10cm 10cm ?
24
Rates of Flow Questions
And what if we were given the time and needed to find one of the lengths? ROUND 3: Mr T builds another T shaped container using two cuboids, with dimensions as pictured. Mr T pours in slime at a constant rate. In 12 hours the water level reaches 25cm from the bottom of the container, as shown. If it takes 21.6 hours to completely fill the tank, determine the height β of the upper cuboid. β 5cm 10cm 10cm Volume filled in 12 hours: ππΓπΓπ +(ππΓππΓπ)=ππππ cm3 Rate at which container fills: ππππ ππ =πππ cm3/hr Total volume: ππΓπΓπ + ππΓππΓπ =πππ+ππππ But also: ππ.πΓπππ=ππππ cm3 β΄πππ+ππππ=ππππ ππππ=ππππ π=ππ cm ? ? 20cm ? 5cm 5cm ? ?
25
Test Your Understanding
Warning: Use consistent units (e.g. metres rather than cm) Volume of water drained: πΓπΓπ.π=π.π m3 Therefore rate per hour: π.πΓπ=π.π m3/hr Total volume: π.π+π.π π Γπ Γπ=π.π m3 Time to fill: π.π π.π =π.ππ hours Therefore time extra to wait: π.ππβπ.π=π.ππ hours or 1 hour, 45 minutes ? ? ? ? ?
26
Exercise 3 ? ? 2 1 65 hours π½=ππΓππ=πππ m3 =πππ πππ π
[Edexcel GCSE(9-1) Mock SetΒ 2Β Spring 2017Β 1F Q21, 1H Q4] The diagram shows a swimming pool. The swimming pool is in the shape of a prism. The swimming pool is filled with water at a rate of 5 litres per second. Jeremy has 10 hours to fill the swimming pool. 1 m3Β = 1000 litres. Will he completely fill the swimming pool in 10 hours? You must show all your working. 2 [Edexcel GCSE Nov2014-2H Q13a] The diagram shows a swimming pool in the shape of a prism. The swimming pool is empty. The swimming pool is filled with water at a constant rate of 50 litres per minute. Work out how long it will take for the swimming pool to be completely full of water. Give your answer in hours. (1 m3Β = 1000 litres) 1 ? π½=ππΓππ=πππ m3 =πππ πππ π Volume filled in 10 hours: ππΓππππΓπ=πππ πππ π Therefore he wonβt fill the pool. ? 65 hours
27
Exercise 3 ? ? 3 4 π= ππ π Volume of water after 37 hours:
[Edexcel GCSE Jun2015-1H Q23] Useful formulae: Volume of a cone with radius π and height β is 1 3 π π 2 β The diagram shows a container for grain. The container is a cylinder on top of a cone. The cylinder has a radius of 3 m and a height ofΒ βΒ m. The cone has a base radius of 3 m and a vertical height of 4 m. The container is empty. The container is then filled with grain at a constant rate. After 5 hours the depth of the grain is 6 metres above the vertex of the cone. After 9 hours the container is full of grain. Work out the value ofΒ β. Give your answer as a fraction in its simplest form. You must show all your working. [Frost] A cone of radius 2cm and height 4cm is gradually filled with water at a constant rate. After 37 hours that water level has risen to 1cm. How long will it take the cone to completely fill with water? 4 4cm 2cm Volume of water after 37 hours: π π π
Γ π π Γπβ π π π
Γ π.π π Γπ = ππ ππ π
Therefore π ππ π
cm3 filled per hour. Total volume = ππ π π
ππ π π
Γ· π ππ π
=ππ hours ? ? π= ππ π
Similar presentations
© 2025 SlidePlayer.com Inc.
All rights reserved.