IB Maths AA 5.6 Notes

Kinematics

Kinematics studies motion using displacement, velocity and acceleration.

• Displacement (s) and distance are typically measured in $\text{m}$.
• Velocity (v) is typically measured in $\text{m s}^{-1}$.
• Acceleration (a) is typically measured in $\text{m s}^{-2}$.
• Time is typically measured in $\text{s}$.

These units come directly from the definitions. Since these are related to time, we can form key derivatives:

$v=\frac{ds}{dt}$

$a=\frac{dv}{dt}=\frac{d^2s}{dt^2}$

Conversely, we can form key integrals.

Displacement from $t_1$ to $t_2$ is

$\int_{t_1}^{t_2}v(t)\,dt$

Distance travelled from $t_1$ to $t_2$ is

$\int_{t_1}^{t_2}|v(t)|\,dt$

It is important to be able to distinguish displacement and distance, and velocity and speed from each other as they are scalar-vector pairs: 

• Displacement $s$ is the straight-line change in position from the starting point. It is a vector quantity, so it can be positive or negative depending on direction.
• Distance travelled is the total path length. It is a scalar quantity, so it is always non-negative.
• Velocity is the rate of change of displacement with respect to time.
• Speed is the magnitude of velocity, so a negative velocity means motion in the negative direction, not negative speed.

$\text{speed}=|v|$

If a particle has velocity $v=-6$, state its speed and direction.

So the particle is moving at speed $6$, but in the negative direction.