Density and Speed Revision Notes for Edexcel GCSE Maths

Density and Speed

Introduction

Density and speed calculations are fundamental concepts in GCSE Mathematics that require you to understand and apply specific formulas. Both topics involve working with relationships between different quantities and require careful attention to units and formula manipulation.

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These concepts form the foundation for more advanced physics and mathematics topics, so mastering the formula manipulation techniques will serve you well in future studies.

Understanding density

Density describes how much mass is packed into a given volume of a substance. Think of it as how "heavy" something feels for its size. A small piece of lead feels much heavier than a same-sized piece of foam because lead has a much higher density.

The relationship between density, mass, and volume can be expressed through three related formulas:

$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$
$\text{Volume} = \frac{\text{Mass}}{\text{Density}}$
$\text{Mass} = \text{Density} \times \text{Volume}$

Common units for density include kilogrammes per cubic metre (kg/m³) and grammes per cubic centimetre (g/cm³).

Using formula triangles for density

Formula triangles provide a visual method for remembering how to rearrange density formulas. The triangle shows mass (M) at the top, with density (D) and volume (V) at the bottom. To find any unknown quantity, simply cover it up and the remaining letters show you what calculation to perform.

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A helpful memory device for remembering the correct order is "DiMoV" (The Russian Agent), which helps you recall that D and V go together at the bottom of the triangle.

Worked example: calculating mass

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When working with density problems, always check that your units are compatible. For instance, if density is given in g/cm³, make sure your volume is in cm³ to get mass in grammes. You can then convert to other units like kilogrammes if needed.

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Worked Example: Calculating Mass from Density and Volume

Consider a chocolate bar with a density of 1.3 g/cm³ and a volume of 1800 cm³. To find the mass:

Step 1: Write down the formula $\text{Mass} = \text{Density} \times \text{Volume}$

Step 2: Substitute the values $\text{Mass} = 1.3 \text{ g/cm³} \times 1800 \text{ cm³}$

Step 3: Calculate $\text{Mass} = 2340 \text{ g}$

Step 4: Convert to kilogrammes $2340 \text{ g} \div 1000 = 2.34 \text{ kg}$

Understanding speed

Speed measures how quickly something moves by comparing the distance travelled to the time taken. It tells you how far an object moves in each unit of time.

The relationship between speed, distance, and time follows three related formulas:

$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$
$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$
$\text{Distance} = \text{Speed} \times \text{Time}$

Common units for speed include kilometres per hour (km/h) and metres per second (m/s).

Using formula triangles for speed

The speed formula triangle works similarly to the density triangle, with distance (D) at the top and speed (S) and time (T) at the bottom. The memory device "SaD Times" helps you remember the correct arrangement.

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The triangle method is particularly useful because it shows you exactly which operation to perform - whether you need to multiply or divide the known quantities.

Worked example: calculating time

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Worked Example: Calculating Time from Distance and Speed

For a car travelling 9 miles at 36 miles per hour:

Step 1: Write down the formula $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

Step 2: Substitute the values $\text{Time} = \frac{9 \text{ miles}}{36 \text{ mph}}$

Step 3: Calculate $\text{Time} = 0.25 \text{ hours}$

Step 4: Convert to minutes $0.25 \text{ hours} \times 60 = 15 \text{ minutes}$

Using formula triangles effectively

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Key Principle: Formula triangles work on a simple principle - cover up the quantity you want to find, and the remaining letters show you the calculation. This method eliminates the need to memorise multiple formula rearrangements.

The key steps are:

Cover the unknown quantity in the triangle
Write down what remains visible
Substitute your known values
Calculate the answer

Remember that the units in your calculation must be compatible. If they don't match, convert them before calculating to avoid errors.

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Key Points to Remember:

Density measures mass per unit volume and is commonly expressed in kg/m³ or g/cm³
Speed measures distance travelled per unit time and is commonly expressed in km/h or m/s
Formula triangles provide a visual method for remembering formula rearrangements
Always check unit compatibility before calculating - convert units if necessary
Use memory devices like "DiMoV" and "SaD Times" to remember the correct triangle arrangements

Density and Speed (Edexcel GCSE Maths): Revision Notes