Straight-line motion with constant acceleration

$$v = v_0 + at$$ $$x = x_0 + v_0t + \frac{1}{2}at^2$$ $$v^2 = v_0^2 + 2a(x-x_0)$$ $$x - x_0 = \frac{v_0 + v}{2}t$$

Straight-line motion with variable acceleration

$$v = v_0 + \int_{0}^{t} a_x dt$$ $$x = x_0 + \int_{0}^{t} v_x dt$$

Freely falling bodies

Projectile motion (2D)

Assuming we have no air resistance, we get: $$a_x = 0\\ a_y = -g$$ $$v_x = v_0\cos\alpha_0\\ v_y = - gt + v_0\sin\alpha_0$$ $$x = (v_0\cos\alpha_0)t + x_0\\ y = -\frac{1}{2}gt^2 + (v_0\sin\alpha_0)t + y_0,$$ where $\alpha_0$ is the angle at which the projectile is launched.

To find the maximum height of the movement, set $v_y = 0$ to find the time $t_1$ at which the projectile reaches its maximum height, and then find $y(t_1)$.

To find the range of the movement, set $y = 0$ to find the time $t_2$ at which the projectile reaches the ground, and then find $x(t_2)$.

Uniform and nonuniform circular motion

$$a_{rad} = \frac{v^2}{R} = \frac{4\pi^2R}{T^2}\\ a_{tan} = \frac{d|\vec{v}|}{dt},$$ where $R$ is the radius of the circular path, and $T$ its period. This gives us $$a = \sqrt{a_{rad}^2 + a_{tan}^2}.$$ In the case of a uniform motion, $a_{tan} = 0$ and thus $a = a_{rad}$.

Weight

$$w = mg$$

Newton's first law

Newton's first law can be stated as: "A body will keep doing what it is doing as long as the sum of the forces acting on it is equal to 0". Or mathematically: $$\sum_{}{}\vec{F} = 0$$

Newton's second law

Newton's second law tells us what happens when the sum of the forces acting on a body is not equal to 0. Mathematically: $$\sum_{}{}\vec{F} = m\vec{a}$$

Newton's third law

Newton's third law states that when a body A exerts a force on another body B, then B will exert a force of equal magnitude in the opposite direction. Or mathematically: $$\vec{F}_{\textrm{A on B}} = -\vec{F}_{\textrm{B on A}}$$

Normal force

The normal force is the force that acts on a body resting on a surface, perpandicular to the surface. The normal force is expressed by: $$\vec{F}_N = m\vec{g}\cos\theta,$$ where $m$ is the body's mass and $\theta$ is the angle at which the surface is. This of course means that for a horizontal surface, $\theta = 0$ and hence $$\vec{F}_N = m\vec{g}$$

Friction

There are 2 types of friction: kinetic friction and static friction. Kinetic friction is the force that acts on a body that slides on a surface, and its magnitude is given by: $$f_k = \mu_kn,$$ where $n$ is the magnitude of the normal force acting on the body, and $\mu_k$ is the friction coefficiant.

Static friction on the other hand is the force that acts on a body that is not moving on a surface. Its magnitude is anywhere between 0 and its maximum, which is given by: $$(f_s)_\textrm{max} = \mu_sn$$ Usually, $$\mu_s \geq \mu_k$$

The direction of the friction's force vector is in the opposite direction of the movement of the body (essentially "slowing" the body).

Air resistance

A decent approximation of the effects of air resistance is given by $$F = kv^2,$$ where $k$ is a constant.

An object in free fall reaches terminal velocity when its air resistance is equal to the gravitational force acting on it, and hence its acceleration becomes $0$. To find the terminal velocity, simply set $a_y = 0$.

Pulleys

In a system with with 1 pulley with 2 masses $m_1$ and $m_2$ on each side of the rope, the tension force of the rope suspending the whole system is given by $$S = 2\frac{m_1m_2}{m_1 + m_2},$$ if we disregard the mass of the pulley itself.

Centripetal force

The centripetal force is the force necessary to keep an object in circular motion. Its value is $$F = m\frac{v^2}{R}.$$

Spring force

The force needed to compress or extend a spring by a distance x is given by $$F = kx,$$ where $k$ is the spring constant. This is referred to as Hooke's law.

Work done by a constant force

A force does work if it produces a displacement. Note that work is a scalar quantity and not a vector. It is defined by: $$ W = \vec{F}\cdot \vec{s} = Fs\cos{\theta},$$ where $\vec{F}$ is the force acting on the object, $\vec{s}$ is the displacement and $\theta$ is the angle between $\vec{F}$ and $\vec{s}$.

Work done by a varying force

When the applied force varies and can be expressed by a function $F_x$, work is given by: $$\int_{x_1}^{x_2} F_x dx$$

Kinetic energy

The kinetic energy of a particle is the amount of work needed to accelerate the particle from rest to a speed $v$. Like work, it is a scalar, and is given by: $$K = \frac{1}{2}mv^2$$

The total amount of work done on a particule is equal to the change of kinetic energy it experiences when going from a speed $v_1$ to a speed $v_2$: $$W_\textrm{tot} = K_2 - K_1$$

Power

Power is the time rate of doing energy. It is a scalar like work and kinetic energy. The average power is given by: $$P = \frac{\Delta W}{\Delta t}$$ The instantaneous power is given by: $$P = \frac{dW}{dt}$$ The rate at which a force $\vec{F}$ does work to make a particle move at velocity $\vec{v}$ is given by: $$P = \vec{F}\cdot\vec{v}$$

Gravitational potential energy

The gravitational potential energy is the energy stored in an object due to its height. It is given by $$E_{pgrav} = mgh,$$ where $m$ is the mass of the object and $h$ its height.

Elastic potential energy

The elastic potential energy is the energy stored in an elastic object due to its stretching or compressing. It is given (in the case of an ideal spring following Hooke's law) by $$E_{pelastic} = \frac{1}{2}kx^2.$$

Mechanical energy

The total mechanical energy is given by $$E_{mec}=E_{pgrav} + E_{pelastic} + K = E_p + K,$$ where $K$ is the object's kinetic energy.

The law of conservation of energy

A force is either conservative or nonconservative. It is nonconservative if its work depends on other factors than kinetic and potential energy, for instance if the path taken affects its work: an example of this is the friction force. However, the following law holds for all forces $$\Delta K + \Delta E_p + \Delta U_{int} = 0,$$ where $U_{int}$ is the internal energy of the body.

Momentum

The momentum of an object refers to the quantity of motion that it has, and is given by $$\vec{p} = m\vec{v}.$$

Impulse

Impulse describes the change in momentum, given by $$J = \Delta p = F \cdot \Delta t,$$ where $F$ is the constant net force acting on the object. If the net force varies with time, we have $$J = \int_{t_1}^{t_2} \sum_{} F dt.$$

Conservation of momentum:

As long as the only forces acting on a system are the forces internal to the system, the total momentum of the system (the sum of all momenta in the system) is constant.

Collision

Center of mass

Rotational kinematics

Given a rigid body rotating about a stationary axis z, then the body's position is described by the angular coordinate $\theta$ and its angular velocity and acceleration are given by $$ \omega_z = \frac{d\theta}{dt}\\ \alpha_z = \frac{d\omega_z}{dt}.$$ If $\alpha_z$ is constant, then the same equations for motion in a straight line apply (although with $\theta$ instead of $x$).

Relating linear and angular kinematics

Given an object travelling at a distance $r$ from the rotation axis, we can derive its speed $v$ and acceleration $a_{tan}, a_{rad}$ from its angular velocity $\omega$ and acceleration $\alpha$ with $$v = r\omega\\ a_{tan} = r\alpha\\ a_{rad}= \omega^2r.$$

Moment of inertia and rotational kinetic energy

The moment of inertia of a body about an axis is a measure of its rotational inertia and is given by $$I = \sum_{i}m_ir_i^2,$$ with $i$ being the particles composing the body. The rotational kinetic energy is given by $$K = \frac{1}{2}I\omega^2.$$

Torque

Torque is a measure of the force that causes an object to acquire angular acceleration. It is given by $$\tau = F \cdot r\sin{\theta},$$ with $r$ being the distance between the axis of rotation $O$ and the point where the force is applied, and $\theta$ the angle between the force vector and $\vec{r}$.

Rotational dynamics

The rotational equivalent of Newton's second law states that $$\sum_{}\tau_z = I\alpha_z.$$

Combined translation and rotation

Work done by a torque

Angular momentum

Rotational dynamics and angular momentum