9709. Mechanics 1. Kinematics in 1 Dimension

Kinematics is the study of motion.

Sometimes we will look at scalar quantities 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

Answers

Acceleration

Acceleration is a vector quantity that measures the time taken of a change in velocity: $a = \frac{v-u}{t}$ ; 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 = $\frac{1}{2}(u+v)$ and so displacement is $s = \frac{1}{2}(u+v)t$

Worked Examples

Exercise

Solutions

Constant Acceleration Formuale

(1.) $v = u + at$

(2.) $s = \frac {u+v}{2}t$

(3.) $v^2 = u^2 + 2as$

(4.) $s = ut + \frac{1}{2}at^2$

(5.) $s = vt - \frac{1}{2}at^2$

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

Worked Examples

Exercise

Answers

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

Answers

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 $a = \frac{v-u}{t}$ ). 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