SL 5.9—Kinematics problems

Kinematics: The branch of mathematics (and physics) that describes the motion of objects, focusing on displacement, velocity, acceleration, and distance , without reference to the forces causing the motion.

In IB Math AA SL, kinematics problems are a direct application of differentiation and integration. The motion of a particle along a straight line is described using three interconnected quantities:

• Displacement $s(t)$ , the position of the particle relative to a fixed origin (a vector quantity)
• Velocity $v(t)$ , the rate of change of displacement (a vector quantity)
• Acceleration $a(t)$ , the rate of change of velocity (a vector quantity)

The three kinematic quantities are linked through calculus. These two relationships are the backbone of every kinematics problem:

Differentiation (going "forward"):
$v(t)=\frac{ds}{dt}a(t)=\frac{dv}{dt}=\frac{d^{2}s}{d{t}^2}$

Integration (going "backward"):
$s(t)=\int v(t)dtv(t)=\int a(t)dt$

You can think of it as a chain:

$s(t)\stackrel{\text{differentiate}}{\to }v(t)\stackrel{\text{differentiate}}{\to }a(t)$
$a(t)\stackrel{\text{integrate}}{\to }v(t)\stackrel{\text{integrate}}{\to }s(t)$