## Math is at the heart of physics. (O Level Maths and Physics Tips)

Studying and practising Mathematics is one of the most useful things an O level student can do.

Not only are the two Maths (E Maths and A Maths) highly intertwined, studying Maths can actually help the students’ Physics too. There are some topics like Vectors and Kinematics in Physics that are also present in Mathematics.

Math is at the heart of physics. So the better your math, the better you’ll do in physics.

A good working knowledge of algebra and trigonometry is needed for Physics.

## E Maths / A Maths: Maximum time per question

Paper 1: 2 hours (120 min) — 80 marks

Max. Time taken per mark: 1.5 min per mark

Paper 2: 2 hours 30 minutes (150 min) — 100 marks

Max. Time taken per mark: 1.5 min per mark

In O Levels Maths, speed and accuracy is very important indeed!

## E Maths Group Tuition Centre; Clementi Town Secondary School Prelim 2012 Solution

Q5) The speed of a boat in still water is 60 km/h.

On a particular day, the speed of the current is $x$ km/h.

(a) Find an expression for the speed of the boat

(I) against the current, [1]

Against the current, the boat would travel slower! This is related to the Chinese proverb, 逆水行舟，不进则退, which means “Like a boat sailing against the current, we must forge ahead or be swept downstream.”

Hence, the speed of the boat is $60-x$ km/h.

(ii) with the current. [1]

$60+x$ km/h

(b) Find an expression for the time required to travel a distance of 80km

(I) against the current,  [1]

Recall that $\displaystyle \text{Time}=\frac{\text{Distance}}{\text{Speed}}$

Hence, the time required is $\displaystyle \frac{80}{60-x}$ h

(ii) with the current. [1]

$\displaystyle \frac{80}{60+x}$ h

(c) If the boat takes 20 minutes longer to travel against the current than it takes to travel with the current, write down an equation in $x$ and show that it can be expressed as $x^2+480x-3600=0$   [2]

Note: We must change 20 minutes into 1/3 hours!

$\frac{80}{60-x}=\frac{1}{3}+\frac{80}{60+x}$

There are many ways to proceed from here, one way is to change the Right Hand Side into common denominator, and then cross-multiply.

$\displaystyle \frac{80}{60-x}=\frac{60+x}{3(60+x)}+\frac{240}{3(60+x)}=\frac{300+x}{3(60+x)}$

Cross-multiply,

$240(60+x)=(300+x)(60-x)$

$14400+240x=18000-300x+60x-x^2$

$x^2+480x-3600=0$ (shown)

(d) Solve this equation, giving your answers correct to 2 decimal places. [2]

$\displaystyle x=\frac{-480\pm\sqrt{480^2-4(1)(-3600)}}{2}=7.386 \text{ or } -487.386$

Answer to 2 d.p. is $x=7.39 \text{ or } -487.39$

(e) Hence, find the time taken, in hours, by the boat to complete a journey of 500 km against the current. [2]

Now we know that the speed of the current is 7.386 km/h.

Hence, the time taken is $\frac{500}{60-7.386}=9.50$ h