Game Development Game Physics Syllabus

How do you normalize a vector, and why is it done in game physics?

Divide by its magnitude: $\hat{v} = \frac{\vec{v}}{|\vec{v}|}$. This produces a unit vector (length 1) that preserves direction, used for directions, normals, and lighting.

What is the derivative of $f(x) = x^{n}$ (power rule)?

$$\frac{d}{dx}x^{n} = n x^{n-1}$$

State the chain rule of differentiation.

$$\frac{d}{dx}f(g(x)) = f'(g(x))\cdot g'(x)$$

Why is differential calculus fundamental to physics simulation?

Motion is described by derivatives: velocity $\vec{v} = \frac{d\vec{r}}{dt}$ and acceleration $\vec{a} = \frac{d\vec{v}}{dt}$. Forces produce accelerations, which are integrated over time to update velocity and position.

State the power rule for integration.

$$\int x^{n}\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$

State the fundamental theorem of calculus.

$$\int_a^b f(x)\,dx = F(b) - F(a)$$ where $F'(x) = f(x)$. Differentiation and integration are inverse operations.

In a physics engine, how are velocity and position obtained from acceleration?

By integration over time: $$\vec{v}(t) = \vec{v}_0 + \int \vec{a}\,dt,\qquad \vec{r}(t) = \vec{r}_0 + \int \vec{v}\,dt$$

What does a matrix represent in linear algebra for game transforms?

A matrix is a rectangular array that encodes a linear transformation (rotation, scale, shear) or, in homogeneous coordinates ($4\times4$), also translation. Applying it to a vector transforms that vector.

Write the standard 2D rotation matrix by angle $\theta$.

$$R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$

What is the identity matrix and its key property?

The identity matrix $I$ has 1s on the diagonal and 0s elsewhere. For any compatible matrix $A$: $AI = IA = A$. It represents the transformation that leaves vectors unchanged.

Why are quaternions preferred over Euler angles for 3D rotation in games?

Quaternions avoid gimbal lock, interpolate smoothly (via slerp), are compact (4 numbers), and compose rotations efficiently without accumulating numerical drift.

Describe the Explicit (Forward) Euler integration method and its formulas.

A first-order method that steps state forward using the current derivative: $$\vec{v}_{n+1} = \vec{v}_n + \vec{a}_n\,\Delta t,\qquad \vec{x}_{n+1} = \vec{x}_n + \vec{v}_n\,\Delta t$$ Simple but can gain energy and become unstable.

How does Semi-implicit (Symplectic) Euler differ from Explicit Euler, and why is it favored?

It updates velocity first, then uses the new velocity for position: $\vec{v}_{n+1} = \vec{v}_n + \vec{a}_n\Delta t$, then $\vec{x}_{n+1} = \vec{x}_n + \vec{v}_{n+1}\Delta t$. It conserves energy better and is more stable for game physics.

What is Verlet integration and one of its advantages?

Position-based scheme: $$\vec{x}_{n+1} = 2\vec{x}_n - \vec{x}_{n-1} + \vec{a}_n\,\Delta t^{2}$$ It is stable, time-reversible, and handles constraints well (used in cloth/ragdoll simulation).

Why is a fixed timestep often used for physics integration?

A constant $\Delta t$ makes simulation deterministic and numerically stable, avoiding artifacts from variable frame rates. Rendering interpolates between fixed physics steps.

Define the moment of inertia and its role in rotational motion.

Moment of inertia $I$ is the rotational analog of mass, quantifying resistance to angular acceleration. For point masses: $I = \sum m_i r_i^{2}$. It relates torque and angular acceleration via $\tau = I\alpha$.

State the rotational form of Newton's second law.

$$\vec{\tau} = I\vec{\alpha}$$ where $\vec{\tau}$ is net torque, $I$ is moment of inertia, and $\vec{\alpha}$ is angular acceleration.

How is torque defined as a vector?

$$\vec{\tau} = \vec{r}\times\vec{F}$$ where $\vec{r}$ is the position vector from the pivot to the force application point. Magnitude: $\tau = rF\sin\theta$.

Define angular momentum for a rigid body.

$$\vec{L} = I\vec{\omega}$$ (for rotation about a principal axis), where $I$ is the moment of inertia and $\vec{\omega}$ is angular velocity. More generally $\vec{L} = \vec{r}\times\vec{p}$.

Give the rotational kinetic energy of a rigid body.

$$KE_{rot} = \tfrac{1}{2}I\omega^{2}$$ A rolling body has total energy $\tfrac{1}{2}mv^{2} + \tfrac{1}{2}I\omega^{2}$.

What distinguishes an elastic collision from an inelastic collision?

In an elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, momentum is conserved but kinetic energy is not (some is lost to heat/deformation). Perfectly inelastic: bodies stick together.

What is the coefficient of restitution $e$, and what do its extreme values mean?

$$e = \frac{|v_{2f} - v_{1f}|}{|v_{1i} - v_{2i}|}$$ It measures bounciness: $e = 1$ is perfectly elastic, $e = 0$ is perfectly inelastic (no bounce).

For a 1D elastic collision between masses $m_1$ and $m_2$, give the final velocity of $m_1$.

$$v_{1f} = \frac{m_1 - m_2}{m_1 + m_2}v_{1i} + \frac{2m_2}{m_1 + m_2}v_{2i}$$

What is an impulse and how does it relate to momentum?

Impulse is the integral of force over time and equals the change in momentum: $$\vec{J} = \int \vec{F}\,dt = \Delta\vec{p} = m\Delta\vec{v}$$ Physics engines resolve collisions by applying impulses.

What is a constraint in a physics engine, and give an example type.

A constraint restricts the relative motion of bodies to satisfy a rule (e.g. a fixed distance). Example: a distance constraint keeps two points a set length apart; joints are constraints connecting rigid bodies.

Name and briefly describe three common joint types used in game physics.

Hinge (revolute): allows rotation about one axis (a door). Ball-and-socket (spherical): allows rotation about all axes (a shoulder). Prismatic (slider): allows translation along one axis only (a piston).

Describe particle kinematics and how a basic particle is updated each frame.

A particle is a point mass with position and velocity but no orientation. Each step: accumulate forces to get $\vec{a} = \vec{F}/m$, then integrate $\vec{v} \mathrel{+}= \vec{a}\Delta t$ and $\vec{x} \mathrel{+}= \vec{v}\Delta t$.

Give the gravitational force between two masses (Newton's law of universal gravitation).

$$F = G\frac{m_1 m_2}{r^{2}}$$ where $G \approx 6.674\times10^{-11}\ \text{N·m}^{2}/\text{kg}^{2}$, directed along the line joining the masses.

Write the formula for kinetic (sliding) friction force.

$$f_k = \mu_k N$$ where $\mu_k$ is the coefficient of kinetic friction and $N$ is the normal force. It opposes the direction of motion.

State Hooke's Law for an ideal spring and the meaning of the sign.

$$\vec{F} = -k\vec{x}$$ The restoring force is proportional to displacement $\vec{x}$ from equilibrium and directed opposite to it; $k$ is the spring stiffness constant.

What is the damping force in a spring-damper system, and what does critical damping mean?

Damping force opposes velocity: $\vec{F}_d = -c\vec{v}$, where $c$ is the damping coefficient. Critical damping is the smallest damping that returns the system to equilibrium without oscillating (fastest non-oscillatory settling).