# UY1: 1-D Kinematics

This is a recap of what you have learned in high school and pre-university. If you require more details, you can visit Speed, velocity & acceleration (O Levels) and/or Kinematics (A Levels).

### Kinematics

Kinematics describe motion in terms of space and time. The types of motion are translational, rotational and vibrational.

Particle approximation: We first assume that the moving object is a point mass, which is valid when the object is small relative to the whole system. (e.g. Earth orbiting around the Sun)

We can describe motion in terms of an equation, a graph or a table.

The motion of a particle is completely known if its position in space is known at all times. (aka. $x \left( t \right)$ for 1-D motion)

A position time graph is a plot of $x \left( t \right)$ against t.

#### Displacement

Displacement is the change in a particle’s position from $x_{i} \left( t_{i} \right)$ to $x_{f} \left( t_{f} \right)$.

$$\Delta x = x_{f}-x_{i}$$

Displacement differs from distance traveled, which depends on the path taken. Displacement can be positive or negative.

#### Speed & Velocity

Average velocity is defined as the ratio of its displacement $\Delta x$ and the time interval $\Delta t$.

\begin{aligned} \left< v_{x} \right> &= \frac{\Delta x}{\Delta t} \\ &= \frac{x_{f}-x_{i}}{t_{f}-t_{i}} \end{aligned}

Average speed is defined as the ratio of total distance traveled to the total time taken.

$$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$$

Velocity is a vector, while speed is a scalar.

Instantaneous velocity is the limiting value of the ratio $\frac{\Delta x}{\Delta t}$ as $\Delta t$ approaches zero.

$$v_{x} = \lim_{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$$

• The above equation is the slope of the position-time graph.
• Instantaneous velocity is also simply referred to as velocity, which is a vector quantity.

Speed (scalar) is given by the magnitude of velocity (vector).

#### Acceleration

When the velocity of a particle changes with time, the particle is said to be accelerating.

Average acceleration is defined as the change in velocity divided by the time interval during which that change occurred.

$$\left< a_{x} \right> = \frac{\Delta v_{x}}{\Delta t} = \frac{v_{x, f}-v_{x, i}}{t_{f}-t_{i}}$$

Instantaneous acceleration equals the derivative of the velocity with respect to time and is given by:

\begin{aligned} a_{x} &= \lim_{\Delta t \rightarrow 0} \frac{\Delta v_{x}}{\Delta t} = \frac{d v_{x}}{dt} \\ a_{x} &= \frac{d v_{x}}{dt} = \frac{d}{dt} \left( \frac{dx}{dt} \right) = \frac{d^{2} x}{dt^{2}} \end{aligned}

Acceleration is the slope of the v-t graph, as velocity is the slope of the x-t graph.

### Kinematics Equations

\begin{aligned} v &= v_{0} + at \\ x &= x_{0} + v_{0} t + \frac{1}{2} a t^{2} \\ v^{2} &= v_{0}^{2} + 2a \left( x-x_{0} \right) \\ x-x_{0} &= \frac{1}{2} \left( v + v_{0} \right) t \end{aligned}

Note: The equations above are only valid for constant acceleration.

### Free Fall

All objects when dropped, fall towards the Earth with nearly constant acceleration, if the height at which the object is dropped is much smaller than the radius of the Earth.

If air resistance can be neglected, then the motion is described as free fall.

Free fall acceleration or acceleration due to gravity:

$$g \approx 9.80 \, \text{m s}^{-2}$$

If an object is in free fall, it will experience an acceleration of magnitude g downwards, regardless of the direction of motion (upwards or downwards).

By convention, the vertical direction, y, is positive upwards. Hence, the acceleration due to gravity is:

$$a =-g$$

Next: 2-D Kinematics

Previous: Basics For Mechanics

Back To Mechanics (UY1)