Compound Measures

• Speed is the rate at which an object covers distance, calculated by dividing the total distance traveled by the time taken.

• The standard formula is $\text{speed} = \frac{\text{distance}}{\text{time}}$, often expressed in units like meters per second ($m/s$) or kilometers per hour ($km/h$).

• Average speed accounts for the total journey, including any stops or changes in pace, whereas instantaneous speed refers to the speed at a specific moment.

• To find distance, the formula is rearranged to $\text{distance} = \text{speed} \times \text{time}$; to find time, it becomes $\text{time} = \frac{\text{distance}}{\text{speed}}$.

• Density measures how much mass is concentrated in a given volume of a substance, serving as a characteristic property of materials.

• The formula is $\text{density} = \frac{\text{mass}}{\text{volume}}$, with common units being grams per cubic centimeter ($g/cm^3$) or kilograms per cubic meter ($kg/m^3$).

• Objects with a higher density than the fluid they are in will sink, while those with a lower density will float, illustrating the practical application of this measure.

• Rearranging the formula allows for the calculation of mass ($\text{mass} = \text{density} \times \text{volume}$) or volume ($\text{volume} = \frac{\text{mass}}{\text{density}}$).

• Pressure is the measure of force applied perpendicular to the surface of an object per unit area over which that force is distributed.

• The formula is $\text{pressure} = \frac{\text{force}}{\text{area}}$, and the standard unit is the Pascal ($Pa$), which is equivalent to one Newton per square meter ($N/m^2$).

• Pressure is inversely proportional to area; for a constant force, increasing the area decreases the pressure (e.g., snowshoes preventing a person from sinking into snow).

• To calculate the force exerted, use $\text{force} = \text{pressure} \times \text{area}$; to find the area, use $\text{area} = \frac{\text{force}}{\text{pressure}}$.

• Unit Consistency: It is critical to ensure all units match before performing calculations. For example, if speed is in $km/h$ and time is in minutes, the time must be converted to hours first.

• Multi-step Conversions: Converting compound units like $km/h$ to $m/s$ involves two steps: converting kilometers to meters (multiply by $1000$) and hours to seconds (divide by $3600$).

Comparison of Compound Measures

• Time Conversion Error: A very common mistake is treating time as a decimal incorrectly. For example, $1$ hour and $30$ minutes is $1.5$ hours, not $1.3$ hours.

• Formula Rearrangement: Always verify the subject of the formula before substituting values. Using the wrong operation (e.g., multiplying instead of dividing) is a frequent source of lost marks.

• Sanity Checks: Evaluate if the final answer is realistic. A human walking at $50 km/h$ or a piece of wood with a density of $20,000 kg/m^3$ suggests a calculation or unit error.

• Area and Volume Units: Remember that $1 m^2 = 10,000 cm^2$ (since $100 \times 100 = 10,000$) and $1 m^3 = 1,000,000 cm^3$. Students often incorrectly use a factor of $100$ for these conversions.