**Kinematics** is the study of motion.

Sometimes we will look at **scalar **quantitie**s** like **distance** and **speed**, and sometimes at **vector quantities** like **displacement** and **velocity**.

What is the difference between a scalar quantity and a vector quantity?

**Speed and Distance**

**s = d ÷ t**, where s = speed, d=distance covered and t = time taken. This applies for constant speed, or gives us the average speed if speed is not constant.

Similarly, average velocity = change in displacement ÷ time taken, which we can rearrange to write as **s = v x t** (s = displacement, v = velocity, t = time)

Clearly units should always be consistent. We will typically use SI units (distances in metres and time in seconds)

**Models and Assumptions**

When we make any **assumptions** (e.g that speed is constant), we should always consider whether the assumption is reasonable in the context of the question (and what the impact of changing the assumption would be).

Throughout this course we will be treating objects as **particles** that are not affected by things such as rotation or deformation.

**Worked Examples**

**Exercise**

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**Acceleration**

Acceleration is a **vector **quantity that measures the time taken of a change in velocity: ; u = initial velocity, v = final velocity.

Positive acceleration represents an increase in velocity over time and negative acceleration reflects a decrease in velocity over time.

If acceleration is constant, then average velocity = and so displacement is

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**Exercise**

**Solutions**

**Constant Acceleration Formuale**

(1.)

(2.)

(3.)

(4.)

(5.)

You should memorise these formulae. which are the most important in mechanics. The first two have been mentioned above and the remaining three you will derive in a following exercise

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**Exercise**

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**Displacement-Time Graphs and Multi-Stage Problems**

On graphs, time is always plotted on the x-axis.

On a **displacement-time (“s-t”) graph**, a horizontal line indicates no movement. In general, the gradient of the line indicates the speed. A curve indicates that there is acceleration. The graph may go below the axis to indicate the opposite direction of motion.

**Worked Examples**

**Exercise**

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**Velocity-time graphs, Speed-time** **graphs and Distance-time graphs**

By contrast, on a **velocity-time** (“v-t”) **graph**, a horizontal line indicates constant speed. The gradient of a velocity-time graph indicates acceleration (think about the formula ). The area below the x-axis indicates negative velocity, which means the direction has changed.

**Important:** The area under a velocity time graph equals **displacement**.

**Discontinuities**

Sometimes velocity can **appear to** change instantaneously, for instance when a bat hits a ball. This will be represented on a v-t graph by a **discontinuity**. We can use a dotted line to mark this.

An s-t graph cannot have a discontinuity (until teleporting is invented 🤔), a bat hitting a ball would be represented by a sharp change in gradient, so the curve would not be smooth.

**Worked Example**

**Various Exercises including Combined on Everything Above**

**Answers**

7.) 4s – Assumptions: e.g. Ball immediately starts moving at 4ms^{-1}, acts as a particle, rebounds with same deceleration.

15. (i.) 0.02ms^{-2}, -0.21ms^{-2}, (ii) 42.5m, (iii) 86.5m