Game plan 0 · Start here
The Motion Map
In most of Specialist, the topic names itself. A complex number question shows you an $i$, and a matrix question shows you a matrix. Motion is the exception. Five different kinds of question all begin “a particle moves…”, and they need different game plans.
Use this page whenever something is moving. Questions 1 and 2 tell you which game plan to open. Question 3 tells you which form of acceleration to use inside Game plan 8. Lines and planes as objects, with equations, intersections and normals, belong to Game plan 3, not here.
Part 1
Three questions
Ask these in order. Stop as soon as you know which game plan to open.
Finding your game plan
- 1Is the motion in a plane or in a straight line? If position is a vector function of time, $\boldsymbol{r}(t)$, or the context is two-dimensional, it is in a plane. Go to Game plan 4 and choose between paths and collisions, projectile motion and circular motion.
- 2In a straight line, does the question name forces: weight, a normal reaction, resistance, equilibrium or momentum? Go to Game plan 8.1. Draw the force diagram, resolve the forces and apply $\Sigma\boldsymbol{F}=m\boldsymbol{a}$.
- 3Is the acceleration given, or can you work it out? Go to Question 3 to choose the form.
Habit
All four forms of acceleration are on the formula sheet, so you never need to remember them. What you need is to choose the right one, and Question 3 does that.
Part 2
Question 3: what are you given, and what does the answer want?
Find the row that matches what the question gives you and what it asks for. Check the last row first. If $a$ is a negative constant times $x$, it is simple harmonic motion and you can stop there.
| You are given | The answer wants | Use |
|---|---|---|
| $x(t)$ or $v(t)$ | anything | Differentiate or integrate directly |
| $a$ is a constant | anything | Integrate directly with respect to $t$ |
| $a=f(t)$ | $v$ or $x$ at a time | $a=\frac{dv}{dt}$, then integrate |
| $a=f(v)$ | a time | $a=\frac{dv}{dt}$, then separate the variables |
| $a=f(v)$ | a displacement | $a=v\frac{dv}{dx}$, then separate the variables |
| $a=f(x)$ | a speed or velocity at a position | $a=\frac{d}{dx}\left(\frac12v^2\right)$, then integrate both sides |
| $a=-\omega^2x$ | anything | Simple harmonic motion. Go to Game plan 8.4 |
After you choose the form
- 1Write the acceleration in the form from the table.
- 2Separate the variables if the table says to, then integrate both sides.
- 3Use the initial conditions to find each constant as soon as it appears.
- 4Rearrange for the quantity the question asks for, and answer in context with units.
Watch for
If you are given $a=f(v)$ and asked for a displacement, $a=\frac{dv}{dt}$ leads you to $t$, which is not what you want. Choose $a=v\frac{dv}{dx}$ so that $x$ appears.
Part 3
Worked routings
Here are four question openings and where the three questions send each one. You do not need to solve them.
“A ball is thrown at 35° to the horizontal…”
Question 1: in a plane. Game plan 4, projectile motion.
“A block rests on a smooth plane inclined at 25°…”
Question 1: a straight line. Question 2: forces are named. Game plan 8.1.
“Its acceleration is $a=-2v$. Find the displacement…”
Question 1: a straight line. Question 2: acceleration. Question 3: $a=f(v)$ and you want a displacement, so use $a=v\frac{dv}{dx}$.
“The particle moves so that $\ddot x=-9x$. Find the period…”
Question 1: a straight line. Question 2: acceleration. Question 3: $a=-\omega^2x$, so it is simple harmonic motion. Game plan 8.4.
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