Mechanics is the study of how and why things move. Before we can understand why, we need a clear way to describe the 'how'. This is the job of kinematics. It’s the vocabulary of motion, focusing on three key ideas: displacement, velocity, and acceleration.

Velocity tells us how fast an object's displacement is changing. It's not just about speed; it also includes direction. A car traveling at 60 miles per hour north has a different velocity than a car traveling 60 miles per hour south.

Acceleration is the rate of change of velocity. Any time an object's speed or direction changes, it is accelerating. Stepping on the gas pedal in a car causes acceleration, but so does hitting the brakes (which is just acceleration in the opposite direction) or turning the steering wheel.

For situations with constant acceleration, a set of simple equations allows us to predict an object's motion. For example, we can find an object's final velocity ($v_f$) if we know its initial velocity ($v_i$), its acceleration ($a$), and the time it was accelerating for ($t$).

We can also calculate the displacement ($\Delta x$) over that time.

Once we can describe motion, we can ask what causes it. The answer lies in the concept of forces, summarized by Isaac Newton’s three laws of motion. These laws form the bedrock of classical mechanics.

Newton's First Law: The Law of Inertia An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. This property of resisting changes in motion is called inertia. A hockey puck sliding on ice will continue to slide for a long time because the force of friction is very small.

Newton's Second Law: Force and Acceleration This law provides the connection between force, mass, and acceleration. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. It's famously summarized in one of physics' most important equations.

Here, $F_{net}$ is the total or 'net' force, $m$ is the mass (a measure of how much 'stuff' an object has), and $a$ is the acceleration. This equation tells us that it takes more force to accelerate a more massive object.

Newton's Third Law: Action and Reaction For every action, there is an equal and opposite reaction. This means that forces always come in pairs. When you push on a wall, the wall pushes back on you with an equal force. When a rocket expels gas downwards, the gas pushes the rocket upwards.

Sometimes, analyzing forces directly can be complicated. The concepts of work and energy give us a powerful alternative way to understand motion. In physics, work has a very specific meaning: it's what happens when a force causes an object to be displaced.

Here, $F$ is the force, $d$ is the displacement, and $\theta$ is the angle between the force and the direction of motion. If you push a box across the floor, you are doing work. If you push on a wall and it doesn't move, you've done no work, no matter how tired you get.

Work is closely tied to energy, which is the capacity to do work. There are many forms of energy, but in mechanics, we often focus on two:

The Work-Energy Theorem states that the net work done on an object equals the change in its kinetic energy. This links the world of forces (through work) to the world of energy.

One of the most fundamental principles in all of science is the Conservation of Energy. It states that energy cannot be created or destroyed, only transformed from one form to another. A rollercoaster at the top of a hill has high potential energy. As it rolls down, that potential energy is converted into kinetic energy, making it move faster.

Finally, power is the rate at which work is done. A powerful engine can do a lot of work in a short amount of time.

What happens when two objects crash into each other? To analyze collisions, we use the concept of momentum. Momentum is often described as "mass in motion" and is calculated by multiplying an object's mass by its velocity.

Like energy, momentum is a conserved quantity. The Law of Conservation of Momentum states that in the absence of external forces, the total momentum of a system remains constant. In a collision between two billiard balls, the total momentum of the two balls before they hit is equal to their total momentum after they hit.

This principle is incredibly useful. It allows us to predict the outcome of collisions without knowing the messy details of the forces involved during the impact.