Year 3 physics: the first test

In Year 3 of high school, physics changes trade. In the first two years it was a two-hours-a-week subject, made of measurements, forces in equilibrium, fluids and heat, with problems that were often solved by plugging numbers into a formula. From Year 3 it becomes three hours and, above all, it becomes quantitative: every problem asks for a declared frame of reference, a split into components and a result with the right unit. At the Italian liceo scientifico (science-focused high school) it is the jump that I see weighing more than the one in maths, and not only because the hours go up: in Year 3 the timetable rises from 27 to 30 hours a week, philosophy arrives, and the school credit towards the final exam starts to count.

The problem is time. The first physics test, in most classes, falls between mid-October and early November, before the student realises that the rules of the game have changed. Whoever gets a 5 on that test almost never has “a gap in the syllabus”: they have applied the Year 2 method to a subject that no longer works that way.

Here you will find what really comes up, the mistakes I correct every year on the same points, a six-step method for setting up a problem, and three exercises worked out in full.

What changes from Year 2 to Year 3

At the liceo scientifico the first part of Year 3, September to October, is almost always about Galilean relativity, frames of reference and apparent forces, and then motion in the plane: projectile motion, circular motion, harmonic motion, the pendulum. From November-December you move on to work and energy, conservative forces, conservation of energy, momentum and collisions. The December test, the one with energy in the presence of friction and with collisions, is the most dreaded of the year. Then, between January and February, gravitation and Kepler’s laws; in spring, perfect gases and thermodynamics.

Look closely at the first line: motion in the plane is built on top of one-dimensional kinematics, which at the liceo scientifico is done between January and March of Year 2 (uniform straight-line motion, uniformly accelerated motion, distance-time and velocity-time graphs, free fall), just before the principles of dynamics. If that part wasn’t already solid by March, it isn’t taken up again calmly in the months of dynamics that close the year, and in Year 3 it is simply taken as known. That is why the first test, formally on motion in the plane, in reality fails those who never consolidated Year 2 kinematics.

In the technical institutes of the technology sector the story is different: physics as a stand-alone subject sits in the first two years (99 hours in Year 1 and in Year 2, partly together with the lab teacher), and from Year 3 the content moves into the specialist subjects: statics, kinematics and dynamics in mechanics; direct current, Kirchhoff and Thevenin in electronics. The method in this article applies there identically: it is the name of the subject in the register that changes, not the way you set up a problem with vectors.

On format there are no surprises compared with the first two years: usually three or four problems plus one or two theory questions in an hour. The difference is that now every problem has two or three steps, and the second depends on the first.

Vectors are the real prerequisite

A large part of the mistakes I correct in the first Year 3 tests are vector mistakes disguised as kinematics mistakes.

A vector has magnitude, direction and sense. Adding two vectors doesn’t mean adding their magnitudes: it means splitting both along the axes, adding the components separately and recombining. With an angle $θ$ measured from the $x$ -axis:

v_{x} = v cos θ v_{y} = v sin θ

and, to go back from the components to the vector,

v = v_{x}^{2} + v_{y}^{2} tan θ = \frac{v _{y}}{v _{x}}

Two consequences that come back everywhere in Year 3. The first: if two vectors are not parallel, the magnitude of the sum is less than the sum of the magnitudes. Whoever writes that a boat going at $4 m/s$ perpendicular to a current of $3 m/s$ has a resultant speed of $7 m/s$ has given the exercise away: it is $5 m/s$ , by Pythagoras.

The second: motions along perpendicular axes are independent. It is the idea that holds up all of projectile motion. A ball rolling off a table and a ball dropped at the same instant from the same height reach the ground together, because the horizontal motion has nothing to say about the vertical one.

Kinematics: the formulas that must be automatic

There are few of them, and you need to know them with the conditions in which they hold, not by heart as strings of symbols.

Uniform straight-line motion, that is constant velocity:

s (t) = s_{0} + v t

Uniformly accelerated straight-line motion, that is constant acceleration:

s (t) = s_{0} + v_{0} t + \frac{1}{2} a t^{2} v (t) = v_{0} + a t

And the relation that skips time, very useful when the problem neither gives nor asks for it:

v^{2} = v_{0}^{2} + 2 a Δ s

Free fall is not a new formula: it is uniformly accelerated motion with $a = - g$ , where $g \approx 9.8 m/ s^{2}$ , once you have chosen the vertical axis positive upwards.

Then there are the two graphs, and this is where the easiest marks of the test are lost.

In the distance-time graph ( $s$ - $t$ ), the slope of the curve is the velocity. A horizontal line is a body at rest, not a body at a constant non-zero velocity. A line with a negative slope is a body coming back.
In the velocity-time graph ( $v$ - $t$ ), the slope is the acceleration and the area under the curve is the displacement. A horizontal line here is a body in uniform motion, not a body at rest. It is exactly the same drawing that in the previous graph meant the opposite.

Reading a $v$ - $t$ graph as if it were an $s$ - $t$ graph is a mistake invisible to the person making it. The question to ask yourself every time, before answering, is just one: what is written on the vertical axis?

The mistakes I see every year in the first test

Not declaring the frame of reference. Two lines at the top of the page (“I take the $x$ -axis horizontal to the right, the $y$ -axis vertical upwards, origin at the launch point”) and half the sign errors disappear. The sign of an acceleration is not a property of the body: it is a consequence of where you pointed the axis.
Getting conversions wrong. $72 km/h$ is not $72 m/s$ : you divide by $3.6$ . It also applies to areas and volumes, where the factor must be raised to a power: from $m^{2}$ to $c m^{2}$ you multiply by $1 0^{4}$ , not by $1 0^{2}$ .
Confusing average speed and instantaneous speed. Average speed is $v_{m} = Δ s /Δ t$ over the whole stretch, and in general it is not the arithmetic mean of the speeds of the individual stretches: it is the most recurrent trap of October, and you will find it worked out in exercise 2.
Misreading the graph, in the two ways seen above: swapping slope and area, or swapping the two types of graph.
Confusing velocity and acceleration in uniform circular motion. The magnitude of the velocity is constant, but the velocity vector changes direction at every instant: so there is an acceleration, directed towards the centre. “Uniform” doesn’t mean “without acceleration”.
Talking about centrifugal force in a problem seen from the ground. In the inertial frame of reference the force acting is the centripetal one, and it points towards the centre. The centrifugal one exists only if you get on board, in a non-inertial frame, which is the distinction the chapter on Galilean relativity, done in September, was there to build.
Substituting numbers too early. Carrying the symbols through and substituting at the end makes a dimensional error visible and earns you marks for the set-up even if you get the last calculation wrong.

On three of these mistakes (force diagrams, signs and dimensional checking) I wrote a separate article: five typical physics mistakes in high school.

The six-step method

The same scheme, on every problem, until it becomes automatic:

Read the whole text and draw. Not an artistic drawing: a sketch with the axes, the body and the vectors that matter.
Write the data with units, already converted to SI units, and write the unknown separately. If you do this well, the method often shows itself.
Declare the frame of reference and, if the motion is in the plane, decide straight away that the problem splits in two: one along $x$ and one along $y$ .
Choose the laws that hold in that specific case, checking the assumptions: is the acceleration constant? does the body start from rest? is friction neglected?
Solve symbolically, substitute at the end.
Check the result on three fronts: do the units work out? is the order of magnitude reasonable (a car doesn’t brake in $0.01 s$ )? does the sign make sense with respect to the axes you chose?

Step 6 is the first to be dropped when time is short, and it is the one that wins back the most marks: thirty seconds of dimensional checking is worth more than thirty seconds on an exercise already written.

Three worked exercises

1. Uniform braking, with the graph

A car is travelling at $72 km/h$ and brakes uniformly until it stops in $5.0 s$ . Calculate the acceleration and the braking distance.

First thing, the conversion: $v_{0} = 72/3.6 = 20 m/s$ . $x$ -axis in the direction of motion, origin where braking begins.

The acceleration from the law of velocity, with $v = 0$ at the end:

a = \frac{v - v _{0}}{Δ t} = \frac{0 - 20 m/s}{5.0 s} = - 4.0 m/ s^{2}

The minus sign is not a mistake to be corrected: it says that the acceleration points backwards with respect to the chosen axis, that is, that the car is slowing down. The braking distance:

s = v_{0} t + \frac{1}{2} a t^{2} = 20 \cdot 5.0 - \frac{1}{2} \cdot 4.0 \cdot 5. 0^{2} = 100 - 50 = 50 m

A cross-check with the graph, which is the reason it is worth drawing: in the $v$ - $t$ plane the braking is a segment going down from $20 m/s$ to zero in $5.0 s$ , and the area underneath is a triangle with base $5.0 s$ and height $20 m/s$ , that is $\frac{1}{2} \cdot 5.0 \cdot 20 = 50 m$ . If in the test you don’t remember the equation of motion, the graph gives it back to you.

2. The average speed that isn’t an average

A student covers $3.0 km$ by bike at $15 km/h$ and the next $3.0 km$ on foot at $5.0 km/h$ . What is the average speed over the whole journey?

The wrong answer, the one that appears in half the tests, is $(15 + 5) /2 = 10 km/h$ . It doesn’t work, because the two stretches last different times and the arithmetic mean weights them equally.

The definition is only one:

v_{m} = \frac{Δ s _{tot}}{Δ t _{tot}}

So you need the times:

t_{1} = \frac{3.0 km}{15 km/h} = 0.20 h t_{2} = \frac{3.0 km}{5.0 km/h} = 0.60 h

v_{m} = \frac{6.0 km}{0.80 h} = 7.5 km/h

The result is closer to the lower speed, and that makes sense: on the slow stretch the student spends three times as long. The arithmetic mean would be right only if the two stretches lasted the same time, not if they were the same length.

3. Projectile motion from a table

A ball leaves a table of height $h = 1.25 m$ with horizontal velocity $v_{0} = 3.0 m/s$ . How far from the edge of the table does it land? With what speed does it reach the ground? Use $g = 9.8 m/ s^{2}$ .

Axes: $x$ horizontal in the direction of motion, $y$ vertical upwards, origin at the edge of the table, at the level of the tabletop. The two motions are independent: uniform along $x$ , uniformly accelerated along $y$ :

x (t) = v_{0} t y (t) = - \frac{1}{2} g t^{2}

The time of flight is decided only by the vertical: the ball lands when $y = - h$ , and its horizontal speed has nothing to do with how long it takes.

h = \frac{1}{2} g t^{2} ⟹ t = \frac{2 h}{g} = \frac{2 \cdot 1.25}{9.8} \approx 0.50 s

Now the range, with the time just found:

x = v_{0} t \approx 3.0 \cdot 0.50 \approx 1.5 m

For the landing speed you need the two components: the horizontal one hasn’t changed, the vertical one has grown. It is better to get $v_{y}$ without going through the rounded value of $t$ , because that number then has to be squared:

v_{y} = - g t = - 2 g h = - 24.5 \approx - 4.95 m/s v_{x} = 3.0 m/s

v = v_{x}^{2} + v_{y}^{2} = 3. 0^{2} + 4.9 5^{2} \approx 5.8 m/s

Whoever adds $3.0$ and $4.95$ directly gets almost $8 m/s$ and loses the mark: the two components are perpendicular, and they combine with Pythagoras.

How to find out where you stand before the test

The quickest way to know which of these points are really yours is not to reread the chapter: it is to try solving. The practice test on motion in the plane is calibrated to the Year 3 programme: kinematics, circular and projectile motion, exactly the themes of this test. If you want to look ahead, there is also the test on work and energy, which is the December topic. The correct answer to each question is free and needs no registration; the step-by-step guided solution unlocks with the first lesson.

What matters is what you do afterwards. Every mistake should go into one of three categories: I didn’t know the theory, I knew the theory but set it up wrongly, I knew everything but got a calculation wrong. They are three different problems, they are cured in three different ways, and telling them apart is more useful than any generic revision. If studying physics feels like a memory exercise, try the Feynman technique applied to physics: explaining a chapter out loud in your own words brings out the points you thought you knew. And if the block in front of the page is not lack of preparation but anxiety, I have also written about how to manage test anxiety in physics and maths.

If you want a hand

On the first Year 3 test the useful work is almost always the same and doesn’t take long: two or three sessions to put vectors, frames of reference and graph reading back in order, then exercises on motion in the plane with the six-step method until it comes by itself. Whoever arrives in October with those three things solid has no more serious problems until energy and collisions in December, a difficult chapter for everyone, but a manageable one.

I work online, one to one; the 10-hour package (rates on the pricing page) is worth it if the road is longer. You will find the full path on the physics tutoring page, and to start from a diagnosis you can book a free 30-minute introductory call: send me the syllabus covered so far and the date of the test, and I will tell you honestly what needs doing between now and then. If instead the failed physics subject has already happened and you want to avoid a repeat, the method for the catch-up exam is in the dedicated article: catching up on failed physics in September.

Year 3 physics: how to prepare for the first test