5.9 KinematicsIB Maths: Analysis and Approaches HL: Revision notes

For motion in a straight line:

• displacement $s$ is the position relative to a fixed point O, with a sign showing which side;
• velocity $v=\frac{ds}{dt}$ is the rate of change of displacement;
• acceleration $a=\frac{dv}{dt}=\frac{d^{2}s}{dt^{2}}$ is the rate of change of velocity.

Example: $s=t^{3}-6t^{2}+9t$ gives $v=3t^{2}-12t+9$ and $a=6t-12$. Units follow: m, m s$^{-1}$, m s$^{-2}$.

Speed is the magnitude of velocity, $|v|$, so it is never negative. If $v(2)=-3$ m s$^{-1}$, the speed is 3 m s$^{-1}$ and the particle is moving in the negative direction.

A particle is at rest when $v=0$. It changes direction only when $v$ changes sign, which usually happens at a moment of rest.

The particle is speeding up when $v$ and $a$ have the same sign and slowing down when they have opposite signs.

The change in displacement between $t_{1}$ and $t_{2}$ is $\int_{t_{1}}^{t_{2}}v(t)\,dt.$ To find the actual displacement from O, add the starting displacement. Alternatively integrate $v(t)$ to get $s(t)+c$ and use a known value (e.g. $s(0)=2$) to find $c$.

Example: $v=8-2t$ from the marker gives $\int_{0}^{6}(8-2t)\,dt=12$ m: the cyclist ends 12 m past the marker.

Distance ignores direction, so $\text{distance}=\int_{t_{1}}^{t_{2}}|v(t)|\,dt.$ Without a calculator: find where $v=0$, split the interval there, integrate each piece and add the absolute values. For $v=8-2t$ on $[0,6]$: $v=0$ at $t=4$, $\int_{0}^{4}=16$, $\int_{4}^{6}=-4$, so distance $=16+4=20$ m while displacement $=12$ m.

For a function like $s=10\cos\frac{\pi t}{4}$ you can also track the positions at each turning point: $10\to-10\to0$ gives $20+10=30$ cm.

IB questions often ask you to describe motion in words. Say where the object is (sign of $s$), which way it is moving (sign of $v$), how fast ($|v|$) and whether it is speeding up or slowing down (compare the signs of $v$ and $a$). For the spring $s=10\cos\frac{\pi t}{4}$, $a=-\frac{\pi^{2}}{16}s$, so acceleration always points towards the rest position: this is the signature of oscillation.