# Subject Content

Study notes for the GCE ‘A’ level syllabus

## 02. Kinematics

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[accordion title=”1. Definitions”]

• Displacement is the distance travelled along a specified direction.
• Speed is the rate of change of distance travelled.
• Velocity is the rate of change of displacement.
• Acceleration is the rate of change of velocity.

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[accordion title=”2. One-Dimensional Motion with Constant Acceleration”]

• $$v=u+at$$
• $$s=(\frac{u+v}{2})t$$
• $$s=ut+\frac{1}{2}at^2$$
• $$v^2=u^2+2as$$

s: displacement
u: initial velocity
v: final velocity
a: acceleration
t: time

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[accordion title=”3. Two-Dimensional Motion”]

• Tip: Sometimes, you will require two equations to solve a kinematics problem. For a parabolic path in a projectile motion without resistive forces, you can draw a table such as the one below and fill in the blank with the information given in the question.
• In the case where a projectile is launched at an angle $$\theta$$ to the horizontal and undergoes the acceleration of free fall, the various horizontal and vertical components of displacement, velocity and acceleration can be expressed in the following way:
 Horizontal Vertical displacement, s $$(u \cos \theta)t$$ $$(u \sin \theta)t+\frac{1}{2}gt^2$$ initial velocity, u $$u \cos \theta$$ $$u \sin \theta$$ initial velocity, v $$u \cos \theta$$ $$u \sin \theta +gt$$ acceleration, a 0 $$g$$ time, t same for both dimensions

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## Base and Derived Quantities

• Physical quantities are classified as base (or fundamental) quantities and derived quantities.
base quantities are chosen to form the base units.
 Base Quantity Base Unit mass kilogram (kg) length metre (m) time second (s) electric current ampere (A) temperature kelvin (K) amount of substance mole (mol) luminous intensity candela (cd)
• Any other physical quantities can be derived from these base quantities. These are called derived quantities.

## Prefixes

• Prefixes are attached to a unit when dealing with very large or very small numbers.
 Power Prefix $10^{-12}$ pico (p) $10^{-9}$ nano (n) $10^{-6}$ micro ($\mu$) $10^{-3}$ milli (m) $10^{-2}$ centi (c) $10^{-1}$ deci (d) $10^3$ kilo (k) $10^6$ mega (M) $10^9$ giga (G) $10^{12}$ tera (T)

## Homogeneity of Units in an Equation

• A physical equation is said to be homogeneous if each of the terms, separated by plus, minus, equality or inequality signs has the same base units.

## Uncertainty

• Absolute uncertainty of a measurement of $x$ can be written as $\Delta x$. This means that true value of the measurement is likely to lie in the range $x-\Delta x$ to $x + \Delta x$.
• Fractional uncertainty = $\dfrac{\Delta x}{x}$
• Percentage uncertainty = $\dfrac{\Delta x}{x}\times100%$
• If the values of two or more quantities such as $a$ and $b$ are measured and then these are combined to determine another quantity $Y$, the absolute or percentage uncertainty of $Y$ can be calculated as follows:
• If $Y = a\pm b$, then  $\Delta Y = \Delta a+\Delta b$
• If $Y = ab$ or $Y = \frac{a}{b}$ , then  $\frac{\Delta Y}{Y} =\frac{\Delta a}{a}+\frac{\Delta b}{b}$
• If $Y = a^n$ then  $\frac{\Delta Y}{Y} = n\frac{\Delta a}{a}$

## Errors

• Systematic errors are errors that, upon repeating the measurement under the same conditions, yield readings with error of same magnitude and sign.
• Random errors are errors that, upon repeating the measurement under the same conditions, yield readings with error of different magnitude and sign.

## Accuracy and Precision

• The accuracy of an experiment is a measure of how close a measured value is to the true value. It is a measure of the correctness of the result.
• The precision of an experiment is a measure of how exact the result is without reference to what that the result means. It is a measure of how reproducible the results are, i.e. it is a measure of how small the uncertainty is.

## Vectors

• A vector quantity has magnitude and direction.
• A scalar quantity has magnitude only.
• Addition of vectors in 2D: $\vec{a}+\vec{b}=\vec{c}$
• Subtraction of vectors in 2D: $\vec{a}-\vec{b}=\vec{d}$
• Methods of finding magnitudes of vectors:
1. resolution of vectors into perpendicular components
2. by scale drawing
3. using:
sine rule: $\frac{a}{\sin \alpha}=\frac{b}{\sin \beta}=\frac{c}{\sin \gamma}$
cosine rule: $a^2 = b^2 + c^2-2bc \cos \alpha$