⊕Read Pelcovits and Farkas (2024) chapter 1 (Background) and review your materials from precalculus with Ms Pepper. These notes cover what you might have done if you tried the summer math problem set, which no one did. It will be mandatory next year. This topic is AP Physics C Mechanics topic 1.1, vectors and scalars, plus some extra that seems useful to review for S&E juniors.

Scalars

A scalar is a number with magnitude but for which direction does not matter.

Temperature, mass, charge, energy, volume, density, viscosityDo not worry if you don’t remember viscosity and kinematic viscosity from chemistry class.

, kinematic viscosity, the amount of a substance in moles, etc are all examples of scalars. The mass of a puppy is simply \(\qty{10}{\kilo\gram}\); it is not \(\qty{10}{\kilo\gram}\) up, \(\qty{10}{\kilo\gram}\) left, right, \(\hat{i}\), \(\hat{x}\), etc. Scalars are the usual numbers we are used to dealing with:

I can add, subtract, multiply, divide scalars and the result is a scalar (closure).
Addition and multiplication are commutative; there’s also an associative property.
When I multiply a sum, I can use a distributive property.
Certain identities hold, like \((x)(1)=x\) and \((x)(0)=0\).
I have to sometimes be careful with zero, most notably, when I divide things.

In physics, scalars may have units associated with them; the puppy has mass \(\qty{10}{\kilo\gram}\); not \(\qty{10}{\meter\squared\per\second}\) and not \(\qty{10}{\kelvin}\) and definitely not just plain 10 with nothing there. Leave units off in physics class and your answer is wrong.

Vectors

A vector is a number with both magnitude and direction.

Position, velocity, acceleration, force, momentum, angular momentum, current, electric field, magnetic field, etc are all examples of vectors. If you were flying an airplane at extremely high speed near the ground, you would care that your altitude is \(\qty{10}{\meter}\) in the positive \(\hat{z}\) direction and that your velocity is \(\qty{100}{\meter\per\second}\) in a horizontal direction; if your altitude was \(\qty{-10}{\meter}\) and your velocity was in the downward direction you would probably be dead because direction matters for a vector.

Vector basics

To distinguish them from ordinary scalars, vectors are often written in bold face (e.g. \(\mathbf{x}\)) or with an arrow on top (\(\vec{a}\)). The latter form is easier to write on the board or by handI will use both methods in class. Your textbook uses the bold face but it is impractical to do bold face when writing by hand.

.

Given a vector \(\mathbf{a} = a_x \hat{i} + a_y \hat{j}\), the magnitude of the vector is given by the Pythagorean theorem:⊕When we square a vector, we take the magnitude squared, a scalar, so that \(\mathbf{v}^2 = v_x^2+v_y^2\). A practical example is kinetic energy, \(KE=\dfrac{1}{2}mv^2\).

\[| \mathbf{a} | = \sqrt{a_x^2 + a_y^2}.\]

The directionA common cause of mistakes is when your calculator is set to degrees or radians and you are working with the other… Be sure to check!

of the vector with respect to the positive \(x\) axis might be obtained from trig: \[\angle \mathbf{a} = \arctan{\dfrac{a_y}{a_x}}\] taking care to choose the correct quadrant, especially when solving problems on a calculator or computer (atan2 or similar in many programming languages). Also check that your device is using degrees or radians as appropriate.

In our notation for the vector \(\vec{b} = b_x \hat{i} + b_y \hat{j}\), \(b_x\) is the \(x\) or horizontal component, \(b_y\) is the \(y\) or vertical component. \(\hat{i}\) and \(\hat{j}\) (or also \(\hat{x}\), \(\hat{y}\), \(\hat{z}\), \(\hat{r}\), \(\hat{\theta}\), etc) are unit basis vectors that point in agreed-uponusually but not always; for example in navigation and vehicle dynamics people use a few different conventions for vehicle-fixed and world-fixed coordinates; East North Up; North East Down; etc. based on what discipline the engineer comes from, that all get very confusing.

directions to allow us to specify our vectors according to agreed-upon coordinate systems. A unit vector is one where its magnitude is 1.

Scalar multiplication

When multiplying a vector by a scalar, the result is a vectorPractical examples: \(\vec{F} = m \vec{a}\), \(\mathbf{p} = m \mathbf{v}\)

:

\[c \mathbf{a} = c a_x \hat{i} + c a_y \hat{j}.\]

Vector addition

When adding two vectors, add the corresponding componentsPractical examples: summing forces \(\Sigma \vec{F} = \vec{F}_1 + \vec{F}_2 + ...\), adding relative velocity.

. For example:

\[\begin{aligned} \vec{s}_1 &= a \hat{i} + b \hat{j} \\ \vec{s}_2 &= c \hat{i} + d \hat{j} \\ \vec{s}_1 + \vec{s}_2 &= (a + c) \hat{i} + (b + d) \hat{j}. \end{aligned}\]

Vector multiplication: dot product

There are two ways to multiply vectors. The dot product or scalar product, written with a dot (\(\cdot\)) gives a scalar result that is the sum of the products of the corresponding componentsPractical example: \(dW = \vec{F}\cdot d\vec{x}\), \(d\Phi = \vec{E}\cdot d\vec{s}\)

:

\[\begin{aligned} \vec{s}_1 &= a \hat{i} + b \hat{j} \\ \vec{s}_2 &= c \hat{i} + d \hat{j} \\ \vec{s}_1 \cdot \vec{s}_2 &= ac + bd. \end{aligned}\]

The dot product has the interesting product that the scalar value of the dot product is the product of the magnitudes times the cosine of the angle between the vectors:

\[\vec{s}_1 \cdot \vec{s}_2 = | \vec{s}_1 | | \vec{s}_2 | \cos{\theta}\]

Refer to your math class materials for proof of this.

Vector multiplication: cross product

The other way to multiply vectors is the cross product or vector product, written with an x (\(\times\)). This operation returns a vectorPractical examples: \(T=\vec{r}\times\vec{F}\), \(F=q\mathbf{v}\times\mathbf{B}\).

.

\[\begin{aligned} \vec{v}_1 &= a \hat{i} + b \hat{j} + c \hat{k} \\ \vec{v}_2 &= d \hat{i} + e \hat{j} + f \hat{k} \\ \vec{v}_1 \times \vec{v}_2 &= (bf-ce)\hat{i} + (cd-af)\hat{j} + (ae-bd)\hat{k}. \end{aligned}\]

The fully 3D cross product is often conveniently remembered as the following determinant, which is easiest evaluated by expanding by minorsIf you don’t know how to do this don’t worry. Yet. We probably won’t get to these for a while.

:

\[\vec{v}_1\times\vec{v}_2 = \det{ \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ a & b & c \\ d & e & f \end{bmatrix} }\]

Since in AP Physics C Mechanics we are usually dealing with 2D vectors in an \(xy\)-plane, things are a bit simplereq. 1 only works for 2D vectors in the \(xy\)-plane!

:

\[\begin{aligned} \vec{U} &= u_x \hat{i} + u_y \hat{j} \\ \vec{V} &= v_x \hat{i} + v_y \hat{j} \\ \vec{U}\times\vec{V} &= (u_x v_y - u_y v_x) \hat{k}. \end{aligned}\qquad{(1)}\]

The cross product operator results in a vector that points in a direction perpendicular to the operands. It is sometimes geometrically interpreted as the area of a parallelogram made by the two operands, pointing in a direction normal to that plane: \[\mathbf{a}\times\mathbf{b} = |\mathbf{a}||\mathbf{b}|\sin{\theta}\hat{n}.\]

Some useful observations regarding vectors

Proof of these is left to you and your math notes from last year.

\[\begin{aligned} 0 \mathbf{v} &= 0 \\ 1 \mathbf{v} &= \mathbf{v} \end{aligned}\]

If \(\vec{a}\) and \(\vec{b}\) are parallel to one another, \[\begin{aligned} |\vec{a}\cdot\vec{b}| &= |\vec{a}||\vec{b}| \\ |\vec{a}\times\vec{b}| &= 0. \end{aligned}\]

If \(\vec{c}\) and \(\vec{d}\) are perpendicular to one another, \[\begin{aligned} |\vec{c}\cdot\vec{d}| &= 0 \\ |\vec{c}\times\vec{d}| &= |\vec{c}||\vec{d}|. \end{aligned}\]

A vector dotted with itself: \[\mathbf{v}\cdot\mathbf{v} = \mathbf{v}^2 = |\mathbf{v}|^2=v^2\]

Parametric equations

An equation or function might be defined parametrically. For example: \[\begin{aligned} x(t) = V_{0,x} t \\ y(t) = V_{0,y} t + \dfrac{1}{2} a_y t^2. \end{aligned}\qquad{(2)}\] \(x(t)\) and \(y(t)\) are both functions of the parameter \(t\).

You could plot the relationship between \(x\) and \(y\) by picking many values of \(t\), obtaining \(x(t)\) and \(y(t)\) for each, and plotting the result with \(x\) along the horizontal axis and \(y\) along the vertical axis. Alternatively, you could make two different plots, one with \(x\) along the vertical axis and \(t\) on the horizontal; and another with \(y\) along the vertical and \(t\) on the horizontal. Both are equivalent and have uses in physics class. For the functions given in eq. 2, plotting \((x,y)\) would give a parabola. Plotting \(x(t)\) versus \(t\) and \(y(t)\) versus \(t\) separately would also give parabolas but they would appear differently.

Similarly, \[\begin{aligned} x(t) = \cos{t} \\ y(t) = \sin{t}. \end{aligned}\qquad{(3)}\]

would appear as a circle when \((x,y)\) is plotted, versus as sines and cosines when \((t,x)\) and \((t,y)\) are plotted. The former might give a picture of the actual trajectory of something through space, while the latter two give additional insight into instantaneous velocity and acceleration.

Other useful math oolies

⊕See https://apcentral.collegeboard.org/media/pdf/ap-physics-c-mechanics-equation-sheet.pdf

Refer to your AP Physics C Mechanics equation sheet for geometry formulas including areas, circumference, perimeter, volume...

SOHCAHTOA

Consider a right triangle with hypoteneuse \(r\) at an angle \(\theta\) from the horizontal. The mnemonic for remembering the relationships between \(\theta\) and the sides is SOHCAHTOA: \[\begin{aligned} \sin{\theta} = \dfrac{\text{opposite}}{\text{hypoteneuse}}\\ \cos{\theta} = \dfrac{\text{adjacent}}{\text{hypoteneuse}}\\ \tan{\theta} = \dfrac{\text{opposite}}{\text{adjacent}} \end{aligned}\] You will apply this in physics for “resolving” or finding components of vectors when given the magnitude.

Also: \[\sin^2{\theta} + \cos^2{\theta} = 1\]

Double angle formulas

\[\begin{aligned} \sin{2\theta} &= 2\sin{\theta}\cos{\theta} \\ \cos{2\theta} &= \cos^2{\theta}-\sin^2{\theta} \\ \tan{2\theta} &= \dfrac{2\tan{\theta}}{1-\tan^2{\theta}}. \end{aligned}\]

Triangle inequality

\[| \mathbf{u} + \mathbf{v} | \leq | \mathbf{u} | + | \mathbf{v} |\]

Logs

\[\log(a b^x) = \log{a} + x \log{b}\]

Solving quadratic equations

You can solve by factoring: \[\begin{aligned} 0 &= V_0 t - \dfrac{1}{2} g t^2 \\ &= t \left( V_0 - \dfrac{1}{2} g t \right). \end{aligned}\]

You can also solve using the quadratic formula: \[\begin{aligned} 0 &= A x^2 + B x + C \\ x &= \dfrac{-B \pm \sqrt{B^2 - 4 A C}}{2 A} \end{aligned}\] It may help you go faster if you know how to use the polynomial solver function on your calculator, which you are allowed to use in physics class. The quadratic formula is not on your AP Physics C Mechanics equation sheet.

Other issues

Units

Use correct units when working problems. S&E students will not be responsible for Mars Lander crashes! More on this next time.

Scientific notation

Be comfortable with scientific notation. It is much easier to write \[M_E = \qty{5.972e24}{\kilo\gram}\] than \[M_E = \qty{5972000000000000000000000}{\kilo\gram}.\] More importantly, you are less prone to make errors in writing the correct number of zeroes; and each such error throws your answer off by at least a factor of ten.

You can speed your work by knowing how to use your calculator’s buttons to quickly enter such quantities. For example, to enter the mass of the earth into a TI-84 you could do 5.972E24 quicker, and with less likelihood of error, than entering (5.972*10^24); and the former does not require you to put parenthesis around it to carry it forward in a larger calculation with correct order of operations.

Your calculator can also be set on NORM, SCI, or ENG mode to help you work quickly. SCI gives you answers in scientific notation, making it easier to copy them down onto your answer sheet. ENG does similar but prefers exponents that multiples of 3 and correspond to kilo-, mega-, giga- etc.

Significant figures

Use correct significant figures when working with measured quantities that properly reflect the uncertainty within a measurement. For example, do not quote a measurement made with a wooden meterstick as \(l = \qty{3.14159265358979323846263383}{\meter}\) because it overstates the precision to which we know \(l\). If we only know the acceleration of gravity to, e.g., 9.8 m s⁻², your answer that was calculated using \(g\) should not have five significant figures.

Sanity checking your answers

Check to see if your answers make sense. Overly large or ridiculously tiny answers may need to be checked. For example, the speed of your labmate rolling on an office chair after accelerating for \(\qty{2}{\second}\) should not be \(\qty{346}{\meter\per\second}\) (speed of sound in air at STP) and should not be \(\qty{3e8}{\meter\per\second}\) (speed of light). When you convert your buddy’s mass from pounds to kilograms to do a momentum calculation, you should not get \(\qty{5.97e24}{\kilo\gram}\), nor should you get \(\qty{9.1e-31}{\kilo\gram}\), and you probably should not get \(\qty{-10}{\gram}\).

Estimation is also a useful skill in science and engineering. Practice trying to bound your answers as to what should be too small, what should be too big, or what is about the right order of magnitude for something. This is less easy to specify than spotting an answer that exceeds the speed of light, but is a good skill to develop.

Navigation links

References

Pelcovits, Robert A, and Joshua Farkas. 2024. Barron’s AP Physics c Premium. Kaplan North America.

Math review