Formulas

Yes, memorizing high-yield formulas is essential for the Physics and Quantitative Reasoning sections of the OAT. [1]

Physics Formulas

MechanicsExa

• Kinematics Equations:
  $v = v_0 + at$
  $d = v_0t + \frac{1}{2}at^2$
  $v^2 = v_0^2 + 2ad$

• Newton's Second Law: $F = ma$

• Kinetic Energy: $KE = \frac{1}{2}mv^2$

• Potential Energy: $PE = mgh$

• Work: $W = Fd \cos(\theta)$

• Power: $P = \frac{W}{t} = Fv$

• Momentum: $p = mv$ [2, 3, 4, 5, 6]

Optics (High Yield for Optometry)

• Snell's Law: $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$

• Thin Lens Equation: $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$

• Magnification: $M = -\frac{d_i}{d_o} = \frac{h_i}{h_o}$

• Lens Power: $P = \frac{1}{f \text{ (in meters)}}$

• Index of Refraction: $n = \frac{c}{v}$

Electricity & Waves

• Ohm's Law: $V = IR$

• Wave Velocity: $v = f\lambda$

• Photon Energy: $E = hf = \frac{hc}{\lambda}$ [7, 8]

Quantitative Reasoning (Math) Formulas

Algebra & Word Problems

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Distance Formula: $d = \text{rate} \times \text{time}$

• Combined Work: $\frac{1}{T_{\text{total}}} = \frac{1}{t_1} + \frac{1}{t_2}$ [9, 10, 11]

Geometry & Coordinate Geometry

• Slope of a Line: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Distance Between Points: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Circle Area: $A = \pi r^2$

• Circle Circumference: $C = 2\pi r$ [12, 13, 14]

Probability & Statistics

• Probability of Event: $P(A) = \frac{\text{Desired Outcomes}}{\text{Total Outcomes}}$

• Permutations (Order matters): $P(n, r) = \frac{n!}{(n-r)!}$

• Combinations (Order doesn't matter): $C(n, r) = \frac{n!}{r!(n-r)!}$ [15, 16, 17, 18]

General Chemistry Formulas (Bonus)

• Ideal Gas Law: $PV = nRT$

• Molarity: $M = \frac{\text{moles of solute}}{\text{liters of solution}}$

• pH Equation: $\text{pH} = -\log[\text{H}^+]$ [19, 20, 21, 22]

Examples:

Here is a sample problem for each of the core high-yield formulas listed above, broken down by section.

Physics: Mechanics

Kinematics ($v = v_0 + at$)

• Problem: A car starts from rest ($v_0 = 0$) and accelerates uniformly at $3 \text{ m/s}^2$ for $5 \text{ seconds}$. What is its final velocity?

• Solution:
  $v = 0 + (3)(5) = 15 \text{ m/s}$

Kinematics ($d = v_0t + \frac{1}{2}at^2$)

• Problem: A ball is dropped from rest off a cliff. If it accelerates down due to gravity ($g \approx 10 \text{ m/s}^2$) for $4 \text{ seconds}$, how far has it fallen?

• Solution:
  $d = (0)(4) + \frac{1}{2}(10)(4)^2 = 5(16) = 80 \text{ meters}$

Kinematics ($v^2 = v_0^2 + 2ad$)

• Problem: An airplane accelerates from rest down a runway at $4 \text{ m/s}^2$. If the runway is $200 \text{ meters}$ long, what is its velocity at takeoff?

• Solution:
  $v^2 = 0^2 + 2(4)(200) = 1600 \implies v = \sqrt{1600} = 40 \text{ m/s}$

Newton's Second Law ($F = ma$)

• Problem: How much net force is required to accelerate a $1200 \text{ kg}$ vehicle at a rate of $2 \text{ m/s}^2$?

• Solution:
  $F = (1200 \text{ kg})(2 \text{ m/s}^2) = 2400 \text{ N}$

Kinetic Energy ($KE = \frac{1}{2}mv^2$)

• Problem: What is the kinetic energy of a $2 \text{ kg}$ projectile moving at a speed of $10 \text{ m/s}$?

• Solution:
  $KE = \frac{1}{2}(2)(10)^2 = 1(100) = 100 \text{ Joules}$

Potential Energy ($PE = mgh$)

• Problem: A $5 \text{ kg}$ textbook is sitting on a shelf $2 \text{ meters}$ above the ground. What is its gravitational potential energy relative to the floor? (Use $g = 10 \text{ m/s}^2$)

• Solution:
  $PE = (5)(10)(2) = 100 \text{ Joules}$

Work ($W = Fd \cos\theta$)

• Problem: A worker pulls a crate along the floor with a force of $50 \text{ N}$ applied at an angle of $60^\circ$ to the horizontal. If the crate moves $10 \text{ meters}$, how much work is done? (Note: $\cos(60^\circ) = 0.5$)

• Solution:
  $W = (50)(10)\cos(60^\circ) = 500(0.5) = 250 \text{ Joules}$

Power ($P = \frac{W}{t}$)

• Problem: An electric motor lifts an elevator by performing $15,000 \text{ Joules}$ of work in $3 \text{ seconds}$. What is the power output of the motor?

• Solution:
  $P = \frac{15,000}{3} = 5,000 \text{ Watts (or 5 kW)}$

Momentum ($p = mv$)

• Problem: What is the momentum of a $0.5 \text{ kg}$ baseball thrown at a velocity of $40 \text{ m/s}$?

• Solution:
  $p = (0.5 \text{ kg})(40 \text{ m/s}) = 20 \text{ kg}\cdot\text{m/s}$

Physics: Optics

Snell's Law ($n_1 \sin\theta_1 = n_2 \sin\theta_2$)

• Problem: A beam of light travels from air ($n_1 = 1.0$) into a plastic block ($n_2 = 2.0$). If the angle of incidence is $30^\circ$, what is the sine of the angle of refraction? (Note: $\sin(30^\circ) = 0.5$)

• Solution:
  $(1.0)(0.5) = (2.0)\sin(\theta_2) \implies \sin(\theta_2) = \frac{0.5}{2.0} = 0.25$

Thin Lens Equation ($\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$)

• Problem: An object is placed $6 \text{ cm}$ ($d_o$) in front of a converging lens with a focal length ($f$) of $2 \text{ cm}$. How far from the lens will the image form ($d_i$)?

• Solution:
  $\frac{1}{2} = \frac{1}{6} + \frac{1}{d_i} \implies \frac{1}{d_i} = \frac{1}{2} - \frac{1}{6} = \frac{3}{6} - \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \implies d_i = 3 \text{ cm}$

Magnification ($M = -\frac{d_i}{d_o}$)

• Problem: An object placed $10 \text{ cm}$ away from a lens produces a real image at a distance of $30 \text{ cm}$ on the opposite side. What is the magnification?

• Solution:
  $M = -\frac{30}{10} = -3 \text{ (The image is inverted and 3 times larger)}$

Lens Power ($P = \frac{1}{f}$)

• Problem: What is the power in diopters of a converging lens that has a focal length of $25 \text{ cm}$?

• Solution: Convert focal length to meters first: $25 \text{ cm} = 0.25 \text{ m}$.
  $P = \frac{1}{0.25} = +4 \text{ Diopters}$

Index of Refraction ($n = \frac{c}{v}$)

• Problem: The speed of light in a certain type of glass is $2.0 \times 10^8 \text{ m/s}$. If the speed of light in a vacuum ($c$) is $3.0 \times 10^8 \text{ m/s}$, what is the index of refraction of the glass?

• Solution:
  $n = \frac{3.0 \times 10^8}{2.0 \times 10^8} = 1.5$

Physics: Electricity & Waves

Ohm's Law ($V = IR$)

• Problem: A circuit has a resistor with a resistance of $5 \ \Omega$ connected to a $12\text{-volt}$ battery. What is the current flowing through the circuit?

• Solution:
  $12 = I(5) \implies I = \frac{12}{5} = 2.4 \text{ Amperes}$

Wave Velocity ($v = f\lambda$)

• Problem: A sound wave has a frequency of $200 \text{ Hz}$ and a wavelength of $1.7 \text{ meters}$. What is the velocity of the sound wave?

• Solution:
  $v = (200 \text{ Hz})(1.7 \text{ m}) = 340 \text{ m/s}$

Photon Energy ($E = hf$)

• Problem: If the frequency of an electromagnetic wave doubles, what happens to the energy of its individual photons?

• Solution: Because $E$ is directly proportional to $f$ ($E = hf$), doubling the frequency perfectly doubles the energy.

Quantitative Reasoning: Math

Quadratic Formula ($x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$)

• Problem: Solve for $x$ in the equation $x^2 - 5x + 6 = 0$.

• Solution: Here $a=1, b=-5, c=6$.
  $x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)} = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2} \implies x = 3 \text{ or } x = 2$

Distance Formula ($d = r \times t$)

• Problem: A cyclist rides at a constant rate of $15 \text{ mph}$ for $3.5 \text{ hours}$. How far do they travel?

• Solution:
  $d = 15 \times 3.5 = 52.5 \text{ miles}$

Combined Work ($\frac{1}{T} = \frac{1}{t_1} + \frac{1}{t_2}$)

• Problem: Pipe A can fill a tank in $3 \text{ hours}$, and Pipe B can fill it in $6 \text{ hours}$. How long will it take to fill the tank if both pipes run at the same time?

• Solution:
  $\frac{1}{T} = \frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \implies T = 2 \text{ hours}$

Slope of a Line ($m = \frac{y_2 - y_1}{x_2 - x_1}$)

• Problem: Find the slope of the line passing through points $(2, 4)$ and $(5, 13)$.

• Solution:
  $m = \frac{13 - 4}{5 - 2} = \frac{9}{3} = 3$

Distance Between Points ($d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$)

• Problem: Find the straight-line distance between coordinate points $(1, 2)$ and $(4, 6)$.

• Solution:
  $d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

Circle Area ($A = \pi r^2$)

• Problem: A circular lens has a radius of $3 \text{ cm}$. What is its area in terms of $\pi$?

• Solution:
  $A = \pi (3)^2 = 9\pi \text{ cm}^2$

Circle Circumference ($C = 2\pi r$)

• Problem: What is the perimeter/circumference of a circle with a diameter of $10 \text{ cm}$?

• Solution: If diameter is $10$, the radius ($r$) is $5$.
  $C = 2\pi(5) = 10\pi \text{ cm}$

Probability ($P = \frac{\text{Want}}{\text{Total}}$)

• Problem: A jar contains $4$ red marbles, $5$ blue marbles, and $1$ green marble. What is the probability of pulling out a red marble at random?

• Solution: Total marbles = $4 + 5 + 1 = 10$.
  $P(\text{Red}) = \frac{4}{10} = \frac{2}{5} \text{ (or 40\%)}$

Permutations ($P(n,r) = \frac{n!}{(n-r)!}$)

• Problem: How many ways can you award 1st, 2nd, and 3rd place trophies to a pool of $5$ racers?

• Solution: Order matters here, so use permutations where $n=5, r=3$.
  $P(5,3) = \frac{5!}{(5-3)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 5 \times 4 \times 3 = 60 \text{ ways}$

Combinations ($C(n,r) = \frac{n!}{r!(n-r)!}$)

• Problem: A clinic wants to choose a committee of $3$ optometrists from a staff of $5$. How many different committees can be made?

• Solution: Order does not matter, so use combinations where $n=5, r=3$.
  $C(5,3) = \frac{5!}{3!(5-3)!} = \frac{5 \times 4 \times 3!}{3! \times 2 \times 1} = \frac{20}{2} = 10 \text{ ways}$

General Chemistry

Ideal Gas Law ($PV = nRT$)

• Problem: If the absolute temperature ($T$) of an ideal gas sample in a rigid container (constant volume $V$) is doubled, what happens to the pressure ($P$)?

• Solution: Since $P$ and $T$ are directly proportional when $V$ and $n$ are constant, doubling $T$ will cause the pressure to double.

Molarity ($M = \frac{\text{moles}}{\text{Liters}}$)

• Problem: A solution is prepared by dissolving $0.5 \text{ moles}$ of $\text{NaCl}$ into enough water to make $2 \text{ Liters}$ of solution. What is the molarity?

• Solution:
  $M = \frac{0.5 \text{ moles}}{2 \text{ L}} = 0.25 \text{ M}$

pH ($\text{pH} = -\log[\text{H}^+]$)

• Problem: What is the pH of an aqueous solution with a hydrogen ion concentration of $[\text{H}^+] = 1.0 \times 10^{-4} \text{ M}$?

• Solution:
  $\text{pH} = -\log(1.0 \times 10^{-4}) = 4$

Would you like to try a few multiple-choice practice questions under timed conditions, or would you like to review test-taking strategies for the real exam?