Kinematics and Dynamics

TL;DR

Kinematics describes how things move using concepts like displacement, velocity, and acceleration, without considering why. Dynamics, on the other hand, explains why things move the way they do by introducing forces and mass. Together, they form the foundation for understanding motion in the universe.

1. The Mental Model

Imagine you're watching a car. Kinematics is just describing its journey: how fast it's going, where it is, and if it's speeding up or slowing down. Dynamics is about why it's moving: the engine pushing it, the brakes stopping it, or gravity pulling it downhill.

2. The Core Material

When we talk about things moving, we first need to define a few concepts.

Displacement vs. Distance

Distance is the total path an object travels, regardless of direction. If you walk 5 meters north and then 5 meters south, you've covered a total distance of 10 meters. Displacement is the straight-line change in position from start to end, including direction. In that same example, if you end up back where you started, your displacement is 0 meters. Displacement is a vector quantity (has magnitude and direction), while distance is a scalar quantity (only magnitude).

Speed vs. Velocity

Similarly, speed is how fast an object is moving (rate of change of distance) and is a scalar. Velocity is how fast an object is moving in a specific direction (rate of change of displacement) and is a vector. Your speedometer shows speed. If you're told a plane is flying at 800 km/h east, that's its velocity.

$$
\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}
$$

$$
\text{Average Velocity} = \frac{\text{Displacement}}{\text{Total Time}}
$$

Acceleration

Acceleration is the rate at which velocity changes. Since velocity is a vector, acceleration can happen in two ways: changing the object's speed, or changing its direction, or both. It's also a vector. If you're in a car and you hit the gas pedal, you accelerate. If you turn the steering wheel while maintaining the same speed, you're also accelerating because your direction is changing.

$$
\text{Acceleration} = \frac{\text{Change in Velocity}}{\text{Time}}
$$

Newton's Laws of Motion (Dynamics)

These three laws explain the relationship between force, mass, and motion.

Newton's First Law (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 external force. Think about a hockey puck on ice; it keeps sliding until friction or a stick acts on it. Inertia is an object's resistance to changes in its state of motion. The more mass an object has, the more inertia it has.

Newton's Second Law

This is where forces really come into play. It states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In simpler terms: more force means more acceleration, but more mass means less acceleration for the same force.

$$
\text{Net Force (F)} = \text{Mass (m)} \times \text{Acceleration (a)}
$$

This is often written as $\text{F} = \text{ma}$. Remember, force and acceleration are vectors, so the net force and the acceleration will always be in the same direction.

Newton's Third Law (Action-Reaction)

For every action, there is an equal and opposite reaction. If you push a wall, the wall pushes back on you with the exact same amount of force. This is why rockets work: they push exhaust gases downward, and the gases push the rocket upward. It's crucial to remember that these action-reaction forces always act on different objects.

3. Worked Example

Let's say a car starts from rest and accelerates uniformly to a velocity of 20 m/s [East] in 5 seconds. If the car has a mass of 1200 kg, what is the net force acting on it?

1. Find the acceleration:

   • Initial velocity ($v_i$) = 0 m/s (starts from rest)
   • Final velocity ($v_f$) = 20 m/s [East]
   • Time ($t$) = 5 s
   • Acceleration ($a$) = $(v_f - v_i) / t$
   • $a = (20 \text{ m/s} - 0 \text{ m/s}) / 5 \text{ s}$
   • $a = 4 \text{ m/s}^2 \text{ [East]}$

2. Find the net force using Newton's Second Law:

   • Mass ($m$) = 1200 kg
   • Acceleration ($a$) = 4 m/s$^2$ [East]
   • Net Force ($F$) = $m \times a$
   • $F = 1200 \text{ kg} \times 4 \text{ m/s}^2$
   • $F = 4800 \text{ N [East]}$

The net force acting on the car is 4800 Newtons [East].

4. Key Takeaways

• Kinematics describes motion (how), while Dynamics explains the causes of motion (why).
• Distance and speed are scalar quantities; displacement and velocity are vector quantities.
• Acceleration is the rate of change of velocity and can be due to changes in speed or direction.
• Newton's First Law defines inertia: objects resist changes in their state of motion.
• Newton's Second Law ($\text{F} = \text{ma}$) quantifies the relationship between force, mass, and acceleration.
• Newton's Third Law states that forces occur in equal and opposite action-reaction pairs acting on different objects.
• The units for force are Newtons (N), where 1 N = 1 kg·m/s$^2$.

5. Now Try It

Imagine you're pushing a 50 kg box across a floor. If you apply a force of 150 N [forward] and the box accelerates at 2 m/s$^2$ [forward], what is the force of friction acting on the box? What success looks like: You'll arrive at a value for the friction force, including its direction, by applying Newton's Second Law and considering the net force.