Algebraic Kinematics & Unit Mismatches
Kinematics problems become simple algebra when you treat units strictly. The fundamental formula is $v = \frac{d}{t}$, where $v$ represents speed, $d$ represents distance, and $t$ represents time.
For example, if you drive a distance of $100 \text{ km}$ ($d$) and it takes you $2 \text{ hours}$ ($t$), your average speed ($v$) is $100 \div 2 = 50 \text{ km/h}$.
The Golden Rule: Units Must Match
If your speed is in km/h, your time MUST be in hours before you multiply or divide. If you are given minutes, convert them to fractional hours immediately ($t \div 60$).
Example: Finding Distance
A car travels at $98 \text{ km/h}$. How far does it travel in exactly $15 \text{ minutes}$?
1. Identify the Mismatch: Speed is in hours, time is in minutes.
2. Convert Time to Hours:
$$ 15 \text{ mins} = \frac{15}{60} = 0.25 \text{ hours} $$3. Calculate Distance ($d = v \times t$):
$$ d = 98 \times 0.25 = 24.5 \text{ km} $$Example: Finding Time
A cyclist travels $15 \text{ km}$ at a speed of $40 \text{ km/h}$. How many minutes does this take?
1. Calculate Time ($t = d \div v$):
$$ t = \frac{15}{40} = 0.375 \text{ hours} $$2. Convert Hours back to Minutes:
To convert from hours to minutes, multiply by $60$.
$$ 0.375 \times 60 = 22.5 \text{ minutes} $$Knowledge Check 1
Marking Schedule:
Q1: Time $= 15/60 = 0.25 \text{ hours}$. Distance $= 120 \times 0.25 = 30 \text{ km}$.
Q2: Time in hours $= 45/60 = 0.75 \text{ hours}$. Minutes $= 0.75 \times 60 = 45 \text{ mins}$.
Proportional Rates & Consumption
Beyond simple distance and time, we often need to calculate the consumption of resources (like fuel or water) over that distance. The standard metric for fuel efficiency is Liters per 100 km (L/100km).
Example: Finding Efficiency
"A 660 km trip consumes 62.7 Liters of petrol. What is the efficiency in Liters per 100 km?"
1. Define the Ratio: We want $\text{Liters}$ divided by $(\text{Hundreds of km})$.
2. Scale the Distance:
$$ 660 \text{ km} = 6.6 \text{ hundreds of km} $$3. Calculate Rate:
$$ \frac{62.7 \text{ Liters}}{6.6 \text{ units}} = 9.5 \text{ L/100km} $$Example: Using Efficiency
"A car operates at an efficiency of 8 L/100km. How many liters are needed for a 250 km journey?"
1. Scale the Distance:
$$ 250 \text{ km} = 2.5 \text{ hundreds of km} $$2. Multiply by Efficiency: Every "hundred" uses $8$ Liters.
$$ 2.5 \times 8 = 20 \text{ Liters} $$Knowledge Check 2
Marking Schedule:
Q1: $400 \text{ km} = 4$ hundreds. Efficiency = $32 \div 4 = 8 \text{ L/100km}$.
Q2: $300 \text{ km} = 3$ hundreds. Liters = $3 \times 7.5 = 22.5 \text{ L}$.
Multi-Step Logistical Problems
In advanced logistical tasks, calculating the "Time" is often a required intermediate step before you can calculate the final financial cost of an operation (such as hourly vehicle rentals or wages).
Target Problem Analysis
"A driver rents a car to travel 150 km at an average speed of 60 km/h. The car rental costs \$15 per hour of driving. The car has a fuel efficiency of 8 L/100km, and fuel costs \$2 per Liter. Find the total financial cost of the trip."
$$ t = \frac{d}{v} = \frac{150 \text{ km}}{60 \text{ km/h}} = 2.5 \text{ hours} $$ $$ \text{Rental Cost} = 2.5 \text{ hours} \times \$15\text{/hr} = \$37.50 $$
$150 \text{ km} = 1.5$ units of $100 \text{ km}$.
$$ 1.5 \times 8 \text{ L/100km} = 12 \text{ Liters used} $$ $$ \text{Fuel Cost} = 12 \text{ Liters} \times \$2\text{/L} = \$24.00 $$
$$ \text{Total} = \$37.50 + \$24.00 = \$61.50 $$
Knowledge Check 3
A driver rents a car for $\$12\text{/hour}$ to travel $200 \text{ km}$ at $80 \text{ km/h}$. The car operates at an efficiency of $6 \text{ L/100km}$, and fuel costs $\$2.50\text{/L}$.
Marking Schedule:
Q1 (Rental Cost): Time $= 200 \div 80 = 2.5 \text{ hours}$. Rental $= 2.5 \times \$12 = \$30$.
Q2 (Total Cost): Fuel Liters $= 2 \times 6 = 12 \text{ L}$. Fuel Cost $= 12 \times \$2.50 = \$30$. Total $= 30 + 30 = \$60$.
Synthesis Task
Combine all skills. Complete 18 mixed-context questions. You have 2 attempts. (Use exact decimals).
Section A: Kinematics & Unit Mismatches
Section B: Proportional Consumption
Section C: Multi-Step Logistics
Section D: Challenge Complex Variations
Student Mastery Report
Take a screenshot of this report to submit to Mr. Shen.