Races and Games; Circular Motion; Surds and Indices

Races and Games

23.1 Key Terminology

Term	Meaning
Race	Contest of speed between two or more
Start	Advantage given to a weaker participant
"A gives B a start of x meters"	B starts x meters ahead of A
"A beats B by x meters"	When A finishes, B is x meters behind
"A beats B by t seconds"	A finishes t seconds before B
"Dead heat"	Both finish at exactly the same time

23.2 Important Formulas

If in a race of 'D' meters, A beats B by 'x' meters:

When A covers D meters, B covers (D - x) meters
Speed ratio: A : B = D : (D - x)

If A gives B a start of x meters in race of D meters:

A runs D, B runs (D - x)
Speed ratio: A : B = D : (D - x)

A's head start formula:

If A beats B by t seconds, B's speed = (D-x)/total_time

23.3 Three-way Race Problems

If A:B = a₁:b₁ and B:C = b₂:c₂:

A:B:C = (a₁ × b₂) : (b₁ × b₂) : (b₁ × c₂)
Simplify by cancelling common factors

Example: A beats B by 25m in 100m race. B beats C by 4m in 100m:

A:B = 100:75
B:C = 100:96
A:C = (100/75) × (100/96) = 100:72
A beats C by 28 meters

Circular Motion

24.1 Key Concepts

Scenario	Meeting Time Formula
Same direction	Circumference / (SA - SB)
Opposite directions	Circumference / (SA + SB)
At starting point again	LCM of their individual lap times

24.2 Meeting at Start vs. Meeting Anywhere

First meeting ANYWHERE (opposite directions):

T = Circumference / (SA + SB)

First meeting ANYWHERE (same direction):

T = Circumference / |SA - SB|

First meeting AT STARTING POINT:

T = LCM(Lap time of A, Lap time of B)

Lap time of X = Circumference / Speed of X

24.3 Example Calculations

Example: P and Q start from same point on 336m circular track. Opposite directions. P: 6 m/s, Q: 8 m/s:

First meeting anywhere = 336/(6+8) = 336/14 = 24 seconds
Lap time P = 336/6 = 56 sec
Lap time Q = 336/8 = 42 sec
Meeting at start = LCM(56, 42) = 168 seconds

Surds and Indices

25.1 Laws of Indices (Exponents)

Law	Formula
Product	aᵐ × aⁿ = aᵐ⁺ⁿ
Quotient	aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power of power	(aᵐ)ⁿ = aᵐⁿ
Zero exponent	a⁰ = 1
Negative exponent	a⁻ⁿ = 1/aⁿ
Fractional exponent	aᵐ/ⁿ = ⁿ√(aᵐ)
Product of same power	aⁿ × bⁿ = (ab)ⁿ
Quotient of same power	aⁿ ÷ bⁿ = (a/b)ⁿ

25.2 Surds

Surd = An irrational number of the form ⁿ√a (where a is rational and the root is irrational)

Types:

Simple surd: √2, √3
Mixed surd: 2√3
Compound surd: √2 + √3

Binomial surd: a + √b

Rationalizing the Denominator:

Multiply and divide by the conjugate
Conjugate of (a + √b) = (a - √b)

Example: 1/(√5 + √3) = (√5 - √3)/[(√5 + √3)(√5 - √3)] = (√5 - √3)/(5-3) = (√5 - √3)/2