Mathematical Requirements

Mathematical Requirements

For AS Physics

Context

In order to be able to develop their skills, knowledge and understanding in physics, and therefore progress on the Physics course, students need to beware of, and to have acquired competence in, the appropriate areas of mathematics relevant to Physics as indicated below.

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1.1Introduction

The following topics need to be covered to successfully meet the demands of the Physics course:

1.1.1Arithmetic and numerical computation

• recognise and use expressions in decimal and standard form
• use ratios and fractions and percentages

1.1.2Handling data

• use an appropriate number of significant figures

1.1.3Algebra

• understand and use appropriate symbols,
• change the subject of an equation,
• solve simple algebraic equations.
• Simplify equations and factorisation
• Quadratic equations

1.1.4Graphs

• translate information between graphical, numerical and algebraic forms
• plot two variables from experimental or other data
• understand that y = mx + c represents a linear relationship
• determine the slope and intercept of a linear graph

2.0Arithmetic and numerical computation

2.1.1Recognise and use expressions in decimal and standard form

Quantities used in Physics are generally in decimal form. Indeed measurements made are in decimal form.

A decimal form refers to the degree to which a measurement is made.

Example 1

The length of a piece of wire is measured as 3.4cm. This means the measurement is to 4 tenths of a centimetre. That is, the instrument measures to one tenth of a millimetre.

Example 2

The time for an object to fall a certain distance is measured as 9.37s. This means the measurement is to 37 hundredths of a second. The instrument measures to one hundredth of a second.

Exercise

Level 1

Write the number of decimal places for the following:

1. 12.3
2. 102.45
3. 1000.7865

Level 2

1. The length of a wire is measured as 1.342m. State how many decimal places this measurement is made and state the degree to which this measurement is made.(Ans: 3dp and one thousandth of a metre. i.e 1 millimetre).
2. A stop watch measures the time of runner as 4.7s. State how many decimal places this measurement is made and state the degree to which this measurement is made.(Ans: 1dp and one tenth of a second).
3. A voltmeter measures the voltage across a resistor as 2.17V. State how many decimal places this measurement is made and state the degree to which this measurement is made.(Ans: 2dp and one hundredth of a Volt).
4. An ammeter measures the current through a resistor as 4.45A. State how many decimal places this measurement is made and state the degree to which this measurement is made.(Ans: 2dp and one hundredth of anAmp).

Level 3

1. A Thermometer measures the temperature of water to be as 150C. State how many decimal places this measurement is made and state the degree to which this measurement is made.(Ans: 0 dp and to the nearest degree).
2. Find 1/3 to one decimal place. (Ans: 0.3).
3. Write 2/3 to two decimal places (Ans: 0.67).
4. Write 15/7 to three decimal places and state the degree to which this calculation is made. (Ans: 2.143).
5. 10911/11 to five decimal places and state the degree to which this calculation is made. ( Ans: 991.90909)

2.1.2Standard Form

Standard form is a process in writing a number in the following form:

Where a is a number between 1 and 9 which is generally a decimal but can be a whole number. b is a whole number. This is expressing a number in terms of powers of 10.

This is used so that large or small numbers can be expressed in a simple form.

Example1

Write 1234 in standard form.

Step 1: Move the decimal point so that number is between 1 and 9. This is 1.234. This is a.

Step 2:The number of decimal points moved to the left is treated as positive and moving to the right as negative. We have in this case +3. This is b.

Step 3: Write in standard form.

Example 2

Write in standard form 0.000569.

Step 1: Move the decimal point so that number is between 1 and 9. This is 5.69. This is a.

Step 2: The number of decimal points moved to the left is treated as positive and moving to the right as negative. We have in this case -4. This is b.

Step 3: Write in standard form.

Exercises

Write in standard form

1. 500.67 (Ans:)
2. 45.3456 (Ans: )
3. 0.00034 (Ans: )
4. -23.4578987 (Ans: )
5. 45 x 34 (Ans: )
6. -0.34 x 500 (Ans: )
7. 100/25 (Ans: )
8. -25/625 (Ans: )
9. (200 x 52)/40 (Ans: )
10. –(500/40) x 7 (Ans: )

2.1.3Ratios and Fractions

Ratios, fractions and percentages are all means of expressing comparisons, this is an usual technique and is used in Physics to express importance of a quantity and usefulness.

Example 1

Two quantities A and B are expressed in the ratio 1:2. What is the relationship of A to B. Hence, if A and B both do a total of 120 J of work how much has A worked and how much has B worked.

The relationship of A to be is such that B is twice the value of A. Thus to find the work of A and B we follow the following steps:

Step 1: Add the ratios. We have 1+2 = 3.

Step 2: Then using this number divide into the total, we have 120/3 = 40.

Step 3: Multiply each of the ratios with this number we have 1x40 =40 and 2x40=80.

Therefore the work done by A is 40J and by B is 80J, We see B has done twice the work.

Example 2

Two quantities A and B are expressed in the ratio 3:2. If B has a value of 40. What is the value of A and find the total shared.

Step 1: The ratio is 3:2. This means that A is 3/2 times bigger than B.

Step 2: Since B has value 40. Then value of A is 3/2 bigger than B. Therefore A has value 40 x 3/2. This gives 60.

Step 3: Thus the total is 60 + 40 = 100.

Exercises

Level 1

1. Divide £800 in the ratio 5:3.
2. Divide £80 in the ratio 4:1
3. Divide £120 in the ration 5:4:3
4. If 7 kiiograms of apples cost £2.80, how much do 12 kilograms of apples cost?
5. If 74 exercise books cost £5.92, how much do 55 exercise books cost?
6. A car travels 205 kilometres on 20 litres of petrol. How much petrol is needed for a journey of 340 kilometres?

Level 2

1. A sum of money is divided into two parts in the ration 5: 7. If the smaller amount is £200, find the larger amount.
2. An alloy consists of copper, Zinc and Tin in the ratio 2:3:5. Find the amount of each metal in 75 kilograms of the alloy.
3. A lone is to be divided into three parts in the ration 2:7:11. If the line is 840millimetres long, calculate the length of each part.

Level 3

1.Two villages have populations 356 and 240 respectively. The two villages are to share a grant of £10728 in proportion to their populations. Calculate how much each village receives.

2. Four friends contribute sums of money to a charitable organisation in the ratio of 2:4:5:7. If the largest amount contributed is £1.40, calculate the total amount contributed by the four people.

2.1.4 Handling data

2.1.4.1 Significant figures

2.1.4.1.1 Rules on significant figures

1. Non-zero numbers (1,2,3,4,5,6,7,8,9) are significant.

2. Zeroes between non-zero numbers are significant.

3. Zeroes which are to the right of the decimal point and at the end of the number are significant.

Examples

34528 is a number written to 5 sf.

108 is a number written to 3 sf.

2341.4006 is a number written to 8 sf.

0.05897 is a number written to 4 sf.

Standard form

A method for determining the number of sf we write the number in standard form.

Examples

34528 = 3.4528 x 105 the number 3.4528 (characteristic) is written to 5 sf.

108 = 1.08 x 102 the characteristic is 1.08 is written as 3 sf.

2341.4006 = 2.3414006 x 103 the characteristic is written to 8 sf.

0.05897 = 5.897 x 10-2 characteristic is 5.897 and is written to 4 sf.

Example

In this example you have to be careful how you write the number it can mean different sf.

3000 can be written as

3000 = 3.000 x 103 this is to 4sf.

3000 = 3.00 x 103 this is to 3sf.

3000 = 3.0 x 103 this is to 2sf.

3000 = 3 x 103 this is to 1sf.

Exercise

How many sf are the following numbers

a) 1234

b) 0.023

c) 890

d) 91010

e) 9010.1

f) 1090.0010

g) 0.00120

h) 3.4 x 104

i) 9.0 x 10-3

j) 9.101 x 10-2

(Ans: a) 4 b) 2 c)2 d) 4 e)5 f) 8 g)3 h)2 i)2 j)4)

General rule

When you add (or subtract) numbers you keep as many decimal places as there are in the least accurate number.

When you multiply (or divide) numbers you keep as many significant digits as there are in the least accurate number.

2.1.5Linear equations

Solve: 2x + 7 = 0.

Answer: In order to solve the equation we isolate the x. This is done as follows:

Step 1: Move the +7 across. As the +7 is moved across the equal sign we change the operation of the +7, it becomes -7. Thus we have,

2x = -7.

Step 2: We now move the 2 across the equal sign. The operation of the 2 changes. It was multiplying it will now divide. Thus we have

x = -7/2 = -3.5.

Solve

Answer: In order to solve the equation we isolate the x. This is done as follows:

Step 1: Move the +2 across. As the +2 is moved across the equal sign we change the operation of the +2, it becomes -2. Thus we have,

,

Thus we have

,

Step 2: We now move the 2x across the equal sign. The operation of the x changes. It wasdividing it will now multiply. Thus we have,

,

Step 3: We now move the 2 across the equal sign. The operation of the 2 changes. It was multiplying it will now divide.

,

Exercises

Level 1

Solve the following equations

1. x+2 = 7
2. t-4=3
3. 3x=9
4. m/3 =4

Level 2

1. 2x+ 5 = 9
2. 3x + 4 = -2
3. 7x+ 12 =5
4. 6x - 3x + 2x = 20
5. 5x - 10 = 3x + 2
6. 1.2x – 0.8 = 0.8x + 12
7. 5(x+2)-3(x – 5) = 29

Level 3

1. Find the number, which when added to the numerator and denominator of the fraction 5/7 gives a new fraction equal to 4/5.
2. Find three consecutive whole numbers whose sum is 48.
3. £380 is divided between A and B. If A receives £144 more than B, how much does each receive?
4. Two numbers when added total 9. 5 times the first number minus 4 times the second number also equals 9. What are the two numbers?

2.1.6Re-arranging equations

Example

Re-arrange the equation to make x the subject of the equation.

Answer: We use theabove method to re-arrange theequation.

Step 1: Move the +3 across. As the +3 is moved across the equal sign we change the operation of the +3, it becomes -3. Thus we have,

,

Step 2: We now move the 2 across the equal sign. The operation of the x changes. It wasmultipying it will now divide. Thus we have,

,

Exercises

Level 1

Re-arrange to make the letter as indicated the subject

1. , for d
2. , for h
3. , for y
4. for T

Level 2

1. , for x
2. , for x
3. D = B – 1.28d, for d
4. , for E

Level 3

1. , for r
2. for T
3. for R
4. for f

2.1.7Factorisation

Example

Answer:In order to simplify the equation through factorisation we need to look for terms that are the same in all the respective quantities.

Step1: Look for the common term in each of the two quantities. This is x.

Step 2: Divide each quantity by the common term. The 2x divide by x is 2. 3x2 divide by x is 3x.

Step 3: Then write the common term x outside a set of brackets and inside the brackets write the results of division from step 2.

This is the result of factorisation.

Exercises

Level 1

Factorise the following

1. 4x +4y
2. 5x – 10
3. 3x2 – 6x
4. 36a2- 9a

Level 2

Factorise the following

1. x2y2 –axy + bxy
2. 5x3 - 10x2y + 15xy2

Level 3

Factorise the following

1. ax+by+bx+ay
2. a2c2 + acd + acd + d2

2.1.8 Quadratic equations

If

Then

Level 1

Questions solve the following

Level 2

1. The area of a rectangle is 6 square metres. If the length is 1 metre longer than the width Find the dimensions of the rectangle.
2. Find the number when added to its square gives a total of 42.
3. Two squares have a total area of 274 square centimetres and the sum of their sides is 88centimetres. Find the side of the larger square.
4. The largest of three consecutive positive numbers is n. The square of this number exceeds the sum of the other two numbers by 28. Find the three numbers.

2.1.9 Graphs

In science we use graphs as means to display data. The graph can be used to deduce relationships between the variables concerned.

Exercises

Level 1

Plot the following graphs

1. The table below gives the temperature at 12.00 noon on seven successive days in June. Plot a graph to illustrate this information.

1. The table below gives corresponding values of x and y. Plot a graph and from it estimate the value of y when x = 1.5 and the value of x when y is 30.

Level 2

1. The areas of circles for various diameters is shown in the table below. Plot a graph with diameter on the horizontal axis and from it estimate the area of a circle whose diameter is 18cm.

1. An electric train starts from A and travels to its next stop 6km from A. The following readings were taken of the time since leaving A ( in minutes) and the distance from A(in km).

Draw a graph of these values taking time horizontally. From the graph estimate the time taken to travel 2km from A.

Level 3

1. The following table gives values of x and y which are connected by an equation of the form y = ax + b. Plot the graph and from it find the values of a and b.

2. In an experiment carried out with a machine the effort E and the load W were found to have the values given in the table below. The equation connecting and W is thought to b of the type E = aw + b. By plotting the grapgh check if this is so and hence find the values of a and b.

3. A car travelling along a straight road passed a post P, and T seconds after passing P its distance(D metres) from P was estimated. The following results were obtained:

Construct another table showing values of T2 against D and verify graphically that T and D are connected approximately by a law of the form,

D = aT2+b

Where a and b are constants.

Use your graph to find

(a) the values of a and b

(b) the time when the car was 175m past P, giving your answer correct to the nearest tenth of a second,

(c) the distance travelled from T = 4 to T = 5.

4.Two variables X and Y are connected by a law of the form

Where a and b are constants. When X = 2.5, Y = 48 and when X=15 Y=529. Draw a graph fo X against and use your graph to estimate values for a and b. Also use your graph to estimate:

(a) the value of X when Y = 441,

(b) the percentage increase in Y as X increase from 6 to 12.

3.0 Trigonometry

3.1 Pythagoras’ theorem

For any right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the remaining sides

We have also the following for a right angled triangle.

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For any triangle we have the general form

Cosine rule

Sine rule

Exercises

Level 1

1. The equal sides of an isosceles triangle are 25cm long and the angle between them is 10°. Calculate the length of the third side. (Ans: 4.358cm)
2. In a triangle ABC, AB = 7cm, BC = 8cm and angle ACB = 53.2°. Calculate angle BAC, given that it is acute. (Ans: 66.25°)

Level 2

1. Calculate the angle between the longer side and diagonal of a rectangle with sides of 4cm and 5cm. (Ans: 38.65°)
2. A ladder restswith its foot on horizontal ground and its top resting against a vertical wall. If the foot of the ladder is 8m from the wall and the ladder makes an angle of 59.3° with the ground, calculate the height of the top of the ladder above ground. (Ans: 13.48m)

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